Let $A$ be a set of $n$ integers. If we write the elements of $A$ in increasing order as $a_1,a_2,\dots,a_n$, then the sequence $a_1+a_1,a_1+a_2,\dots,a_1+a_n,a_2+a_n,\dots,a_n+a_n$ is strictly increasing, and therefore the sumset $A+A$ has size at least $2n-1$. It is an easy exercise to prove that equality holds if and only if $A$ is an arithmetic progression. In the other direction, since $a_i+a_j=a_j+a_i$ for every $i$ and $j$, $A+A$ has size at most $n(n+1)/2$, and equality holds for any suitably dissociated set: for example, it holds if $a_i=3^i$ for each $i$. Erdős and Szemerédi noted that one could use appropriate mixtures of these two extreme constructions to show that all sumset sizes between $2n-1$ and $n(n+1)/2$ could be achieved, an observation that led Nathanson to ask what happens for higher sumsets. That is, he defined $\mathcal{R}(h,k)$ to be the set of all possible values of $|hA|$, where $A$ is a set of $k$ integers and $hA$ denotes the $h$-fold sumset of $A$, and he asked what $\mathcal{R}(h,k)$ is for general $h$ and $k$. The analogues of the two extreme bounds just mentioned for $h=2$ are $hk-h+1$ and $\binom{h+k-1}{h}$, again achieved by arithmetic progressions at one end and dissociated sets at the other. However, what goes on in between is more subtle, since it is _not_ true that every cardinality in between can be achieved. In particular, Tang-Xing and Schinina independently identified an interval of missing cardinalities for each $h,k\geq 3$, and Tang-Xing identified a second and third interval. This paper shows that there is a sequence of missing intervals $I_1,\dots,I_r$ that form a Freiman-homomorphic image of a triangle, in the sense that the left end-points and right end-points of the $I_j$ form arithmetic progressions and the lengths of the $I_j$ decrease from $r$ to $1$, and the author conjectures that every missing cardinality belongs to one of these intervals whenever $k>h$. (The statement is not true in general if $k\leq h$.) The main result of the paper is that for each fixed $h$, this conjecture holds for sufficiently large $k$. As when $h=2$, the proof works by combining dense and sparse sets, but the way this is done is far subtler and less obvious than it is when $h=2$. As well as asking about the possible cardinalities of $hA$, one can also ask about the sets that achieve those cardinalities. For example, define $N(h,k)$ to be the smallest integer $N$ such that every cardinality in $\mathcal R(h,k)$ can be achieved by a set $A$ of diameter at most $N$. What can one say about the dependence of $N(h,k)$ on $h$ and $k$? A consequence of the results in the paper is that for each fixed $h$, $N(h,k)$ grows at most exponentially in $k$, a result that was previously obtained by different methods by Nathanson. This was recently improved to a polynomial dependence by ChatGPT 5.5 Pro, making heavy use of the ideas in this paper but adding some new ideas of its own. More details about ChatGPT's result can be found [in a blog post](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#more-6666) co-written by the author and Timothy Gowers (who suggested the problem to ChatGPT).
We introduce a notion of \emph{efficient stability} for finite presentations of groups. Informally, a finite presentation using generators $S$ and relations $R$ is \emph{stable} if any map from $S$ to unitaries that approximately satisfies the relations (in the tracial norm) is close to the restriction of a representation of $G$ to the subset $S$. This notion and variants thereof have been extensively studied in recent years, in part motivated by connections to property testing in computer science. The novelty in our work is the focus on \emph{efficiency}, which, informally, places an onus on small presentations -- in the sense of encoding length. The goal in this setup is to achieve non-trivial tradeoffs between the presentation length and its modulus of stability. With this goal in mind we analyze various natural examples of presentations. We provide a general method for constructing presentations of $\mathbb{Z}_2^k$ from linear error-correcting codes. We observe that the resulting presentation has a weak form of stability exactly when the code is \emph{testable}. This raises the question of whether testable codes give rise to genuinely stable presentations using this method. While we cannot show that this is the case in general, we leverage recent results in the study of non-local games in quantum information theory (Ji et al., Discrete Analysis 2021) to show that a specific instantiation of our construction, based on the Reed-Muller family of codes, leads to a stable presentation of $\mathbb{Z}_2^k$ of size polylog$(k)$ only. As an application, we combine this result with recent work of de la Salle (arXiv:2204.07084) to re-derive the quantum low-degree test of Natarajan and Vidick (IEEE FOCS'18), which is a key building block in the recent refutation of Connes' Embedding Problem via complexity theory (Ji et al., arXiv:2001.04383).
We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$ with $(A+B)\cup (AB)$ monochromatic, answering a question from a recent paper of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated version of this result that also generalizes a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for fields. Our main new technique is an extension of the color focusing method that involves trees rather than sequences.
Denote by ${\mathfrak s}({\mathbb F}_p^d)$ the minimal number $s$ such that among any $s$ (not necessarily distinct) vectors in ${\mathbb F}_p^d$ one can find $p$ vectors whose sum is zero. Denote by ${\mathfrak w}({\mathbb F}_p^d)$ the weak Erd\H{o}s-Ginzburg-Ziv constant, that is, the maximal number of vectors $v_1, \ldots, v_s \in {\mathbb F}_p^d$ such that for any non-negative integers $\alpha_1, \ldots, \alpha_s$ whose sum is $p$ we have $\alpha_1 v_1 + \ldots + \alpha_s v_s = 0$ if and only if $\alpha_i = p$ for some $i$. We show that for any $p$ and $d$ we have an upper bound ${\mathfrak w}({\mathbb F}_p^d) \le {2d-1 \choose d}+1$. The main result of this paper is that for any fixed $d$ and $p \rightarrow \infty$ we have an asymptotic formula ${\mathfrak s}({\mathbb F}_p^d) \sim {\mathfrak w}({\mathbb F}_p^d) p$. Together with the upper bound on ${\mathfrak w}({\mathbb F}_p^d)$ this result in particular implies that ${\mathfrak s}({\mathbb F}_p^d) \le 4^d p$ for all sufficiently large $p$. In order to prove the main result, we develop a framework of convex flags which generalize usual polytopes in many ways. Many classical results of Convex Geometry translate naturally to this new setting. In particular, we obtain analogues of Helly Theorem and of Central Point Theorem. Also we prove a generalization of Integer Helly Theorem of Doignon. One of the main tools in our argument is the Flag Decomposition Lemma which asserts that for any subset $X \subset {\mathbb F}_p^d$ one can find a convex flag which approximates $X$ in a certain way. Then, Integer Central Point Theorem and other tools allow us to solve the problem for this approximation. Finally, in order to lift the solution back to the original set $X$ we apply the Set Expansion method of Alon-Dubiner.
We show that there exist Z(2) symbolic systems that are strongly irreducible and have no (fully) periodic points.
We show that if K >= 1 is a parameter and S is a finite symmetric subset of a group containing the identity such S-2n <= K S-n for some integer n >= 2K(2), then S-3n <= exp(exp(O(K-2))) S-n . Such a result was previously known only under the stronger assumption that S2n+1 <= KSn . We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.
Color avoidance for monotone paths, Discrete Analysis 2025:23, 14 pp. The famous Erdős–Szekeres theorem states that every sequence $a_1,\dots,a_{rs+1}$ of $rs+1$ real (and without loss of generality distinct) numbers has an increasing subsequence of length $r+1$ or a decreasing subsequence of length $s+1$. One way to prove it is to colour each pair of integers $mn$ with $ma_n$. This gives us a red/blue colouring of the complete graph $K_{rs+1}$. Defining a path $m_1,\dots,m_k$ to be _monotone_ if $m_1<\dots0$, but the problem is still wide open. This paper concerns generalizations of some of these ideas to hypergraphs. For this we first need to say what we mean by a "path" in a $k$-uniform hypergraph. The two commonest definitions are of _loose paths_, where consecutive edges intersect in exactly one vertex, and _tight paths_, where consecutive edges intersect in $k-1$ vertices. When $k=2$, these definitions coincide, but for $k>2$ they are very different. In this paper tight paths are considered. More precisely, the authors look at _tight monotone paths_, which are paths where one has a monotone sequence $v_1r/2$, then one needs not a $(k-2)$-fold iterated exponential but a $(k-3)$-fold iterated exponential, applied to a suitable power of $n$ that depends on $r$ and $s$ only. Note that when $k=3$ this implies that we do not need any exponentials, and indeed the first thing the authors do is prove that $A_3(n;3,2)$ is at most $n^9$. The paper concludes with some appealing open problems. The most obvious is whether the upper bound provided in this paper can be matched by a lower bound, at least for the number of exponentials needed.
In this paper, we study in prime fields the exceptional set estimates, which can be viewed as a refinement of Marstrand's orthogonal projection theorem. Additionally, we address a Furstenberg-type problem, which is closely related. It is shown that the two-dimensional result implies all higher-dimensional results in the prime-field setting.
Irreducibility and periodicity in $\mathbb{Z}^{2}$ symbolic systems, Discrete Analysis 2025:17, 37 pp. One of the central aims of symbolic dynamics is to understand how local rules governing configurations lead to global dynamical behaviour. In the one-dimensional case, this picture is fairly well understood. Strongly irreducible (SI) systems, which can be thought of as models with a robust mixing property, form a particularly important class. Roughly speaking, strong irreducibility means that any two finite patterns that appear in the system can always be combined into a single configuration, provided they are placed far enough apart. A classical theorem, proved independently by Bertrand and by Weiss, shows that such systems necessarily contain dense periodic points. Thus, in one dimension SI systems exhibit both mixing-type behaviour and periodic structure. In two dimensions the situation is more delicate. Lightwood proved that SI shifts of finite type (those defined by finitely many forbidden patterns) always contain periodic points, suggesting that perhaps the one-dimensional theory continued to hold. But for general $\mathbb{Z}^2$-subshifts the question remained open: does strong irreducibility alone force periodicity? This paper gives a decisive answer. Hochman constructs a strongly irreducible $\mathbb{Z}^2$-subshift that has no periodic points at all. The construction is intricate, since even small perturbations tend to introduce periodicity, but the broad idea is to build in many local “witnesses” of aperiodicity (small patterns that by themselves already prevent global periodicity) and distribute them in large geometric patterns in such a way that they can always be combined without conflict. The outcome is a system that retains the mixing property of strong irreducibility but avoids global periodic structure altogether. The result closes a long-standing problem and highlights a striking contrast between one and two dimensions: properties that are tightly linked in one setting can pull apart in the other. It also brings new techniques to the subject, combining combinatorial and geometric ideas that may have further applications. The paper concludes with a number of open questions, the most significant of which asks whether strongly irreducible $\mathbb{Z}^3$ shifts of finite type must contain periodic points. Another direction is to ask whether strongly irreducible sofic shifts (factors of finite-type shifts) in two dimensions can have no periodic points. Finding such examples would represent a major breakthrough.
Small doubling implies small tripling at large scales, Discrete Analysis 2025:9, 9 pp. In additive combinatorics, the well-known Plünnecke--Ruzsa inequalities state that if $A$ is a finite nonempty subset of an abelian group satisfying $|2A| \le K|A|$ then $|m A - n A| \le K^{m+n} |A|$ for all nonnegative integers $m, n$. In particular it follows that the small-doubling hypothesis $|2A| \le K|A|$ is sufficient to guarantee control of all iterated sum or difference sets. In the case of nonabelian groups this has long been known not to hold. For example if $A$ is the union of a finite subgroup $H \le G$ and another element $g$ then $A^2 = Hg \cup gH \cup \{g^2\}$ has size comparable to $A$, but $A^3$ contains the subset $HgH$, which may be as large as $|H|^2$ if $H \cap H^g = \{e\}$. However, it does still follow from a version of the Ruzsa triangle inequality that the small-tripling condition $|A^3| \le K|A|$ implies $|A^{\epsilon_1} \cdots A^{\epsilon_m}| \le K^{3(m-2)} |A|$ for all $\epsilon_i = \pm 1$, so it has become common in nonabelian additive combinatorics and the theory of growth in groups to assume a small-tripling condition. It turns out that things are better if $A$ is taken to be a ball $B(r)$ (with respect to some generating set) for some large radius $r$, which is a common and important special case. In this case, a result of Breuillard and Tointon shows that the small-tripling condition can almost be weakened to a small-doubling condition. Precisely, if $|B(2r+1)| \le K |B(r)|$ then $|B(3r)| \le O_K(|B(r)|)$. This paper goes even further and shows that, if $r$ is sufficiently large depending on $K$, then in fact the true small-doubling hypothesis $|B(2r)| \le K |B(r)|$ is indeed already sufficient to imply $|B(3r)| \le O_K(1) |B(r)|$. Variants of this result are also given for balls in locally compact groups and in locally finite vertex-transitive graphs. Since there are several results in the literature assuming a small-tripling condition for balls, the results in this paper allow for the weakening of hypotheses in several previous results in the structure theory of groups and graphs, such as results of Easo and Hutchcroft on uniform finite presentation of groups of polynomial growth (see for instance [their paper published recently in Discrete Analysis](https://discreteanalysisjournal.com/article/127778-uniform-finite-presentation-for-groups-of-polynomial-growth)) and a quantitative version of the theorem of Trofimov on vertex-transitive graphs of polynomial growth. The argument is short and clever, involving a combination of combinatorial and analytic techniques, including multiplicative energy and covering lemmas. The bounds obtained are effective and explicit.
A very sharp threshold for first order logic distinguishability of random graphs, Discrete Analysis 2025:6, 17 pp. Let $G_1$ and $G_2$ be two independent random graphs with $n$ vertices. The probability that they are isomorphic is very small, but they nevertheless share many important properties: to give just one of many possible examples, with high probability they will contain roughly the same number of triangles. There are certainly _some_ properties that they have a significant probability of not sharing. For instance, with probability 1/2, one of them will have an even number of edges and the other an odd number of edges. However, this property of graphs is quite complicated in that it depends on every single edge (or more precisely, for each pair $xy$ of vertices the property depends on whether $xy$ is an edge). Are there properties that depend on far fewer variables but that can still distinguish between $G_1$ and $G_2$ with reasonable probability? The usual way to formulate this question precisely is to ask whether there is a property $P$ given by a first-order formula in the language of graphs (roughly speaking, that means that the atomic formulae can use equality and the "is an edge" relation, and the formula as a whole is built up from atomic formulae using connectives and quantification) that depends on only a small number of variables, with the property that the probability that $P(G_1)$ holds and $P(G_2)$ does not hold is bounded away from 0. Because first-order quantification yields statements such that they have to hold for all vertices or for at least one vertex, it is hard to avoid properties $P$ that are true with probability close to 0 or 1. For example, the property $$\forall x\ \exists y\ \exists z\ xy\in E(G)\wedge yz\in E(G)\wedge xz\notin E(G),$$ which states that every vertex is an end vertex of an induced path of length 2, holds with probability $1-o(1)$. To see this, note that for each $x,y,z$, the probability that $yz\in E(G)$ and $xz\notin E(G)$ is 1/4, so for each $x$ and $y$, the probability that there exists $z$ with $yz\in E(G)$ and $xz\notin E(G)$ is $1-(3/4)^{n-2}$. Similarly, for each $x$, the probability that there exists $y$ with $xy\in E(G)$ is $1-(1/2)^{n-1}$. Since these probabilities are exponentially close to 1, the probability that such $y$ and $z$ exist for _every_ $x$ is also exponentially close to 1. A development of this argument can be used to show that every fixed first-order formula satisfies a zero-one law, in the sense that it holds with probability $o(1)$ or $1-o(1)$. This paper concerns the point at which the argument breaks down, in the following sense. If the formula is not fixed, but instead the number of variables is allowed to depend on $n$, how many variables are needed for a property $P$ such that the probability that $P(G_1)$ holds and $P(G_2)$ does not hold is bounded away from 0, or, better still, is $1/4-o(1)$? It is relatively straightforward to see that the bound one would expect here is logarithmic. In one direction, if $k$ is the number of variables, then in order for probabilities such as $(1-2^{-k})^n$ not to be exponentially small, we need $2^{-k}$ to be comparable to $1/n$, so we need $k$ to be at least logarithmic. And in the other direction, the value of $k$ for which the formula $$\exists x_1,\dots,x_k\ \bigwedge_{i\ne j} (x_ix_j\in E(G))$$ holds with probability approximately 1/2 is logarithmic in $n$. (The formula of course states that $G$ contains a clique of size $k$.) The paper proves a much more precise result than this, obtaining the right bound for $k$ up to an additive constant. In fact, it narrows down the right value of $k$ to just four integers, namely $\hat k+1, \hat k+2, \hat k+3$ and $\hat k+4$, where $\hat k$ is roughly the integer part of $\log_2n-2\log_2(\log n)+\log_2(\log 2)$. Previously, the best known upper bound for general $n$ was of the form $\hat k+O(\log\log n)$, though in 2003 Kim, Pikhurko, Spencer and Verbitsky obtained an upper bound of the form $\hat k+O(1)$ that applies to infinitely many $n$. The fact that $k$ must be at least $\hat k+1$ follows from a simple argument that is similar to the argument for induced $P_2$s above, so the content of this result is the upper bound. To obtain an upper bound, one must find a formula in $k$ variables that holds with probability close to 1/2. (The authors are aiming for the stronger statement where the probability of the event $P(G_1)\wedge\neg P(G_2)$ is close to 1/4.) The proof makes heavy use of the second-moment method. It is delicate, because the construction of the formula depends in an important way on whether $\log_2n-2\log_2(\log n)+\log_2(\log 2)$ is or is not close to an integer. In the simplest case, the authors define a certain monotone decreasing family $\mathcal F$ of graphs, and $P(G)$ holds if and only if $G$ has an induced subgraph $H$ that is isomorphic to a graph in $\mathcal F$, together with a vertex $x$ that is joined to all the vertices in $H$.
Power saving for the Brown-Erdős-Sós problem, Discrete Analysis 2025:5, 16 pp. It has long been known that there are important connections between extremal questions concerning hypergraphs and extremal questions in additive combinatorics. This realization dates back at least as far as the famous $(6,3)$-theorem of Ruzsa and Szemerédi, which has several equivalent formulations, including the equally famous triangle removal lemma. In its hypergraph formulation, the theorem states that if a 3-uniform hypergraph has $n$ vertices and no six of those vertices span three edges, then there are $o(n^2)$ edges. (Here an "edge" means a triple of vertices that belongs to the hypergraph.) A moment's reflection shows that if the hypergraph is linear -- that is, no two edges overlap in more than one vertex -- then the only way that six vertices can span three edges is for those three edges to form what is sometimes called a "triforce", which is a configuration of the form $xyu, yzv, xzw$. Thus, another way to state the theorem is to say that for every $\delta>0$ there exists $n$ such that every linear hypergraph with $n$ vertices and at least $\delta n^2$ edges (note that the number of edges of a linear hypergraph is trivially at most $\binom n2$) contains a triforce. To see how this is relevant to additive combinatorics, let $G$ be an Abelian group of odd order $n$ and let $A$ be a subset of $G$ of density $\delta$. Form a 6-partite linear hypergraph $H$ with vertex sets $X,Y,Z,U,V,W$ that are each copies of $G$, and let $xyu$ be an edge if $y-x=u$ and $u\in A$, let $yzv$ be an edge if $z-y=v$ and $v\in A$, and let $xzw$ be an edge if $z-x=2w$ and $w\in A$. Then $H$ has positive density (independent of $n$) so for large enough $n$ it must contain a triforce. It is a small exercise to check that that triforce must contain exactly one vertex in each vertex set, which therefore gives us three elements $x,y,z$ of $G$ such that $y-x$ and $z-y$ belong to $A$ and $z-x$ belongs to $2.A$ (the dilate of $A$ by a factor of 2). From this we obtain $u,v,w\in A$ with $2w=u+v$ -- that is, an arithmetic progression. This is not quite a proof, because the arithmetic progression can be degenerate, but a slightly more careful version of the argument yields that the number of triforces exceeds the number of degenerate arithmetic progressions. Motivated by this result, it is natural to formulate the following general problem: how many edges can a 3-uniform hypergraph on $n$ vertices have if no $v$ vertices span $e$ or more edges? In particular, the well-known Brown-Erdős-Sós question asks whether if $e$ is fixed and $v=e+3$, then this number is $o(n^2)$. A well-known result in this direction, due to Sárközy and Selkow in 2004, is that the number of edges is $o(n^2)$ if $v=e+\lfloor \log_2 e\rfloor+2$. Note that when $e=3$, $\lfloor\log_2e\rfloor=1$, so this result generalizes the Ruzsa-Szemerédi $(6,3)$-theorem. This result has been improved more recently: in 2017 Solymosi and Solymosi obtained the conclusion if $e=10$ and $v=14$, instead of the 15 that would be given by the Sárközy-Selkow theorem, and Conlon, Gishboliner, Levanzov and Shapira obtained the first asymptotic improvement by showing that the conclusion holds if $v=e+\lceil 26\log e/\log\log e\rceil$. This paper concerns a related question: how large does $v$ have to be (as a function of $e$) for the bound to be not just $o(n^2)$ but $O(n^{2-c})$ for some positive constant $c$ (depending on $e$)? The main result is that it suffices if $v=\lfloor \log_2 e\rfloor +38$. Thus, their bound matches the Sárközy-Selkow bound (but not the Conlon-Gishboliner-Levanzov-Shapira bound) up to an additive constant. A bound of $e+2\log_2e+C$ was previously known, but quite a lot easier to prove and therefore not formally published anywhere -- it could perhaps be described as folklore -- but to remove the factor of 2 the authors use tools such as the sunflower lemma that had not previously made an appearance in this circle of ideas. One reason to be interested in power savings in the Brown-Erdős-Sós problem is an observation of Gowers and Long that a power saving for the $(9,5)$ problem would yield a positive solution to the following stubbornly open problem in additive combinatorics. **Problem.** _Do there exist constants $c, C>0$ such that for every $n$, every subset of $\{1,2,\dots,n\}$ of size at least $Cn^{1-c}$ contains distinct elements $x,y,z,w$ such that $2(x+y)=z+3w$?_ A straightforward modification of the proof of Roth's theorem yields a bound of $o(n^2)$ for this problem, but what makes the question interesting is that in the other direction it is _not_ possible to modify the Behrend construction to give a lower bound of the form $n^{1-o(1)}$.
A dimension-free Remez-type inequality on the polytorus, Discrete Analysis 2025:4, 21 pp. The classical Remez inequality is a powerful tool in approximation theory. It allows one to bound the maximum value of a real polynomial $p$ on a closed interval $\Delta$ in $\mathbb R$ in terms of its supremum on an arbitrary subset $J$ of $\Delta$ of positive measure. More precisely, if $m$ stands for Lebesgue measure, the inequality states that the maximum of $p$ on $\Delta$ is at most $(4m(\Delta)/m(J))^d\sup_J|p|$, where $d$ is the degree of $p$. Recently, motivated by important applications to various problems in modern analysis, there has been considerable interest in generalizing the classical Remez inequality to a multivariate setting, that is, replacing the interval by certain compact sets in higher dimensions. The existing literature in this direction mainly deals with convex sets, and the resulting multivariate generalizations of the Remez inequality typically involve constants that depend on dimension. In general this dependence is necessary, but that does not preclude the possibility that for some interesting classes of sets and subsets one might obtain dimension-independent constants. The main result of this paper is a dimension-free Remez-type inequality for analytic polynomials on the polytorus $\mathbb T^n$. The subsets in question are of the form $\Omega_K^n$, where $\Omega_K$ is the cyclic group of $K$th roots of unity. Let $\alpha$ be a non-negative sequence in $\mathbb Z^n$. We write $|\alpha|$ for $\sum_{i=1}^n\alpha_i$, and if $z\in\mathbb C^n$, we write $z^\alpha$ for $\prod_{i=1}^nz_i^{\alpha_i}$. We also write $[K]$ for $\{0,1,\dots,K-1\}$. Then an analytic polynomial of degree at most $d$ and individual degree at most $K-1$ (where $K\geq 2$) means a function $f$ of the form $\sum_{\alpha\in[K]^n,|\alpha|\leq d}c_\alpha z^{\alpha}$. The dimension-free Remez inequality proved in this paper states that if $f$ is an analytic polynomial of degree $d$ and individual degree at most $K-1$, then $\|f\|_{\mathbb T^n}\leq C(d, K)\|f\|_{\Omega_K^n}$, where we write $\|f\|_X$ to denote the supremum norm of the restriction of $f$ to $X$. The key point of interest here is that the constant $C(d,K)$ depends only on $d$ and $K$, and not on $n$. This is a noteworthy new result obtained by means of quite complex and original arguments, combining several subtle analytical ingredients. The authors use this result to provide a new proof of the Bohnenblust-Hille inequality for functions on products of cyclic groups. The original form of the Bohnenblust-Hille inequality states that if $f$ is a polynomial of degree at most $d$ defined on $\mathbb T^n$, then $\|\hat f\|_{\frac{2d}{d+1}}\leq C(d)\|f\|_\infty$. Note that the Plancherel identity immediately gives the inequality $\|\hat f\|_2=\|f\|_2\leq\|f\|_\infty$, and since the $\ell_p$ norms on $\mathbb Z^n$ increase as $p$ decreases, the Bohnenblust-Hille inequality is stating that for degree-$d$ polynomials one can obtain a significant improvement on the trivial bound. A similar inequality was proved by the authors when $f$ is defined on the group $\Omega_K^n$, where now the constant depends on $K$ as well. The main theorem of this paper yields a different proof of this result. The results of this paper have subsequently been improved by the authors, together with Lars Becker and Ohad Klein. However, the proof here is quite different, and interesting in its own right.
Square packings and rectifiable doubling measures, Discrete Analysis 2025:3, 40 pp. A measure $\mu$ on a metric space is $m$-rectifiable if it assigns full measure to a countable collection of Lipschitz images of bounded subsets of $\mathbb{R}^m$. Measures that are 1-rectifiable are a major object of study in geometric measure theory, and their structure is now well understood. Much less is known when $m > 1$. For example, a simple metric characterization of Lipschitz curves has been known since the 1920s, but no such result is available for higher-dimensional Lipschitz images. This article undertakes a study of the $m>1$ case, with interesting and intriguing results. First, the authors obtain several sufficient conditions for a measure to be $m$-rectifiable for general $m$. Of particular interest is the criterion given in Theorem 2.5, with a general and robust construction of a Lipschitz map based on a geometric observation concerning the packing of axis-parallel cubes in $\mathbb{R}^n$. The authors then use that criterion to prove their main result: given two integers $m,n$ such that $2\leq m0$, called a doubling constant, such that for all $r>0$ and for all $x$ in the support of $\mu$, $$ 0<\mu(B(x,2r))\leq D\mu(B(x,r))<\infty, $$ where $B(x,r)$ is the ball of radius $r$ centred at $x$. For $m=1$, this was already known. Specifically, Garnett, Killip, and Schul (2010) proved that for any $n\geq 2$ there exists a doubling measure $\mu$ on $\mathbb{R}^n$ and a rectifiable curve $\Gamma\subset\mathbb{R}^n$ such that $\mu(\Gamma)>0$. This is already a highly counterintuitive result: for example, it would be false if $\Gamma$ were required to be even slightly smoother, and the restriction of $\mu$ to $\Gamma$ must be singular with respect to the 1-dimensional Hausdorff measure on $\Gamma$. The $m\geq 2$ case is much more difficult, requiring the new methods introduced here. Additionally, the authors are able to impose additional conditions on the dimensionality of $\mu$. For example, their result implies that there exist doubling measures $\mu$ on $\mathbb{R}^3$ supported on sets of Hausdorff dimension 0.0001 and packing dimension 1.9999 that are 2-rectifiable and purely 1-unrectifiable. Both the results and the methods developed here mark significant progress in our understanding of $m$-rectifiable measures for general $m$, and are likely to inspire further work.
A note on the power sums of the number of Fibonacci partitions, Discrete Analysis 2025:2, 13 pp. This article concerns the number of ways of writing a positive integer $n$ as a sum of strictly increasing Fibonacci numbers. Denote this number by $r_F(n)$. The function $r_F$ is sometimes called the _Fibonacci partition function_. The Fibonacci partition function behaves in quite a strange way: [https://oeis.org/A000119](it appears in OEIS) (the Online Encyclopaedia of Integer Sequences) as sequence A000119, and starts 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 3, 1, 3, 3, 2, 4, 2, 3, 3. Various authors have found recursive definitions of it and some have even found formulae that are explicit enough to make it possible to calculate it efficiently. In particular, Sam Chow and Tom Slattery found a way to describe $r_F(n)$ in terms of how $n$ decomposes as a sum of non-consecutive Fibonacci numbers (it is an easy exercise to show that there is exactly one way of doing this for each $n$, which is known as the _Zeckendorf expansion_ of $n$). One simple fact that sheds a little light on the sequence is that if $F_n$ is the $n$th Fibonacci number, then $r_F(F_n-1)=1$. To see this, note that for every $m$, the sum $1+2+3+\dots+F_m$ is equal to $F_{m+1}+F_{m-1}+F_{m-3}+\dots$ (with the exact form of the end of the sum depending on the parity of $m$), which one can show by induction to be strictly less than $F_{m+2}-1$. It follows that in any representation of $F_n-1$, we must use $F_{n-1}$, but since $F_n-1=F_{n-1}+(F_{n-2}-1)$, we then find by induction that the only representation of $F_n-1$ is $F_{n-1}+F_{n-3}+\dots$. (For example, $20=13+5+2$ and $33=21+8+3+1$.) Paul Stockmeyer also showed that $r_F(n)\leq\sqrt{n+1}$, with equality if and only if $n=F_m^2$ for some positive integer $m$. When a number-theoretic function on the positive integers is far from smooth, one way to understand it better is to smooth it out by looking at sums or averages. Chow and Slattery used their formula to determine the asymptotic growth rate of the sum $\sum_{n=1}^Nr_F(n)$, showing that the sum is within a constant factor of $N^{\log 2/\log\phi}$. Here, $\phi$ is the golden ratio $(\sqrt 5+1)/2$, the presence of which in the formula is clearly to be expected. More surprisingly, the limit of $N^{-\log 2/\log\phi}\sum_{n=1}^Nr_F(n)$ does not exist. Chow and Slattery also suggested that it would be interesting to look at higher moments. In a later paper, Chow and Owen Jones made a start on this by determining the rate of growth of $\sum_{n=1}^Nr_F(n)^2$, which turns out to be $N^{\log\lambda_1/\log\phi}$, where $\lambda_1$ is the largest root of the cubic $x^3-2x^2-2x+2$. This result was interesting, because it was a higher power than the obvious lower bound one obtains from the estimate for $\sum_{n=1}^Nr_F(n)$ combined with the Cauchy-Schwarz inequality. This paper, using quite different methods, tackles all the moments. That is, it determines the growth rate of $\sum_{n=1}^Nr_F(n)^p$ for every positive integer $p$. The basic approach is to build on a construction by Berstel of a deterministic finite automaton that accepts a pair $(x,y)$ of 01 sequences if and only if $y$ starts with a 1 and has no pair of consecutive 1s, and the Fibonacci sums corresponding to $x$ and $y$ are equal (where we let the earlier bits represent larger Fibonacci numbers). Thus, the number of pairs of length up to $n$ accepted by the automaton counts the sum over all pairs $(x,y)$ where $y$ represents the Zeckendorf expansion of some integer less than $F_n$ and $x$ represents some expansion of that integer. A deterministic finite automaton is a directed graph where each vertex represents a _state_, one of which is the _initial state_ and some subset of which are designated as _accepting states_. The edges are labelled with the elements of some alphabet. Then a sequence of letters from that alphabet is fed in and one walks around the graph, starting at the initial state and then for each term in the sequence moving along the edge labelled by that term. Here the sequence consists of pairs $(x_i,y_i)$, which belong to the alphabet $\{0,1\}^2$. A sequence is said to be _accepted_ if at the end of this walk one has landed on an accepting state. One can obtain asymptotics for the number of accepted sequences by thinking of the directed graph as a transition matrix and finding its largest eigenvalue. If this is $\lambda$, then the number of accepted sequences of length $\ell$ is asymptotic to $\lambda^\ell$. If one wishes to use a similar method to determine the asymptotics for $\sum_{n=1}^Nr_F(n)^p$, then one is counting sequences $(x^{(1)},\dots,x^{(p)},y)$ such that each $(x^{(j)},y)$ is accepted by Berstel's automaton. The obvious way to attempt to do this is to take $p$ copies of Berstel's automaton and run them in parallel with the same $y$ and different $x$s. However, if one does that, then most of the states of the resulting automaton cannot be reached, whereas to apply the method for counting accepted states one needs a graph that is strongly connected (meaning that there is a directed path from each vertex to each other vertex). The main work of this paper is to understand the automaton that results when one throws away the inaccessible states, and to understand it well enough to prove that it is strongly connected and to determine the largest eigenvalue $\lambda_p$ of the transition matrix. This eigenvalue is not given in closed form. Rather, what emerges from the analysis is an efficient algorithm that can compute, for each $p$, the minimal polynomial of $\lambda_p$, which is a monic polynomial with integer coefficients and degree $p$ if $p$ is odd and $p+1$ if $p$ is even. The paper also includes a proof that $\lambda_p^{1/p}$ converges to $\sqrt\phi$, so the growth rate of $\lambda_p$ is also now understood.
Uniform finite presentation for groups of polynomial growth, Discrete Analysis 2025:1, 29 pp. Let $G$ be a finitely generated infinite group with generators $a_1,\dots,a_k$. The _ball of radius_ $r$ in $G$ is defined to be the set of all elements of $G$ that can be written as words of length at most $r$ in $a_1,\dots,a_k$ and their inverses, and the _growth rate_ of $G$ with respect to $a_1,\dots,a_k$ is defined to be the function that takes $r$ to the size of the ball of radius $r$. For example, the growth rate of the free Abelian group on $k$ generators is roughly $(2r)^k/k!$ (with respect to those generators), and the growth rate of the free group on $k$ generators is $2k+2k(2k-1)+2k(2k-1)^2+\dots+2k(2k-1)^{r-1}$. It can be shown without too much difficulty that the growth rate does not depend very strongly on the generating set chosen: for instance, if the growth rate is within a constant factor of a polynomial for one set of generators, it will be within a constant factor of the same polynomial for any other set of generators. In particular, the following notion is well defined: a group is of _polynomial growth_ if for some (and hence every) set of generators the growth rate is bounded above by some polynomial. A famous theorem of Gromov, proved in 1981, states that a finitely generated group is of polynomial growth if and only if it is virtually nilpotent, which means that it has a nilpotent subgroup of finite index. (The "if" part of this statement is not trivial, but it is much easier, and was proved in 1968.) A consequence of this theorem is that groups of polynomial growth are finitely presented -- that is, they have a presentation with a finite number of relations. Note that saying that a group $G$ with generators $x_1,\dots,x_k$ is finitely presented is equivalent to saying that there exists $n$ such that the set of relations of length at most $n$ that hold among $x_1,\dots,x_k$ gives a presentation for $G$. This paper is interested in a quantitative version of the fact that groups of polynomial growth are finitely presented. As such, it fits into a general programme of obtaining quantitative versions of results connected with Gromov's theorem, of which a famous example is a theorem of Breuillard, Green and Tao, which yields a quantitative version of Gromov's theorem itself. What does this mean? Well, if $B_r$ is the ball of radius $r$ and $|B_r|$ grows like a polynomial of degree $d$, then $|B_{Cr}|$ will be around $C^d|B_r|$. It turns out that a convenient quantitative notion of polynomial growth is that $|B_{3r}|\leq K|B_r|$ for some sufficiently large $r$. We shall not state the Breuillard-Green-Tao theorem here (it is stated in the paper on page 2), but in broad terms it implies that for every $K$ there exist $r_0$ and $C$ such that if $|B_{3r}|\leq K|B_r|$ for some $r\geq r_0$, then $|B_{mr}|\leq m^C|B_r|$ for every $m$, and the group satisfies a quantitative version of the virtual nilpotency property. This paper does something similar with the property of being finitely presented. Let $G$ be a group with a symmetric set of generators $x_1,\dots,x_k$ and let $R_s$ be the set of words in $x_1,\dots,x_k$ of length at most $s$ that yield the identity in $G$. Then as commented above, the statement that $G$ is finitely presented is the statement that for some $s$ the words in $R_s$ give a presentation of $G$, or equivalently that $F_k/\langle\langle R_s\rangle\rangle\cong G$, where $F_k$ is the free group on $x_1,\dots,x_k$ and $\langle\langle R_s\rangle\rangle$ is the normal subgroup of $F_k$ generated by $R_s$. This in turn is equivalent to the statement that the sequence of normal subgroups $\langle\langle R_s\rangle\rangle$ is eventually constant. The main result of this paper is to make this statement uniform in a certain sense. The statement is that for every $K$ and $k$ there exist $r_0$ and $C$ such that if $G$ has $k$ generators and $|B_{3r}|\leq K|B_r|$ for some $r\geq r_0$, then there are at most $C$ integers $n\geq\log_2r$ for which $\langle\langle R_{2^{n+1}}\rangle\rangle\ne\langle\langle R_{2^n}\rangle\rangle$. That is, if we say that $G$ _has a new relation on scale_ $n$ if $\langle\langle R_{2^{n+1}}\rangle\rangle\ne\langle\langle R_{2^n}\rangle\rangle$, then beyond $r_0$ there are at most $C$ scales on which $G$ has a new relation. This result has been strengthened in [a subsequent paper of Romain Tessera and Matthew Tointon](https://arxiv.org/abs/2403.02485). However, the proof in this paper is considerably shorter. One might think of asking for more, namely an upper bound for the scale on which the last new relation appears. However, as the authors point out, simple examples show that that is too much to ask for. The example they give is $\prod_{i=1}^k\mathbb Z/n_i\mathbb Z$, with the obvious generators, which has a new relation on scale $\log_2 n_i$ (give or take integer parts) for each $i$. Note that the _number_ of scales at which new relations appear is $k$, so very nicely bounded in terms of $k$ and $K$ (which is around $3^k$), but nothing can be said about the sizes of those scales. The main result is interesting in itself, but it is proved with a major application in mind, known as Schramm's locality conjecture, which the authors prove in another paper. This conjecture concerns percolation in vertex-transitive graphs, and states the following. Let $G$ be an infinite connected graph and let $p\in[0,1]$. Now choose a random subset of the edges of $G$, by picking each edge with probability $p$, with all choices independent. We are then interested in whether the resulting graph contains an infinite component. The _critical probability_ of $G$ is defined to be the infimum over all $p$ such that there is an infinite component with probability 1. Typically, one studies vertex-transitive graphs (that is, graphs for which the automorphism group acts transitively on the vertices), in which case there is an infinite component with probability 1 if and only if for every vertex $x$ there is a positive probability that $x$ is contained in an infinite component. Schramm conjectured that the critical probability should depend only on the "local structure" of $G$, in the following sense. Let $(G_n)$ be a sequence of vertex transitive graphs, each with critical probability less than 1, that converges to a vertex-transitive connected graph $G$. Then the critical probabilities of the graphs $G_n$ converge to the critical probability of $G$. Here the notion of convergence for the graphs is that for every $r$ there exists $n_0$ such that for every $n\geq n_0$ the ball of radius $r$ in $G_n$ is isomorphic to the ball of radius $r$ in $G$. Not surprisingly, there are close connections between groups of polynomial growth and vertex-transitive graphs of polynomial growth (this just means that the sizes of the balls around each vertex grow polynomially). The proof by the authors of Schramm's locality conjecture involves a delicate splitting into cases according to the growth rate of the graph $G$ -- surprisingly, their two cases are whether or not $G$ has _quasi_-polynomial growth -- and makes important use of the uniformity obtained in this paper.
Using the recent proof of the polynomial Freiman-Ruzsa conjecture over F-p(n) by Gowers, Green, Manners, and Tao, we prove a version of the polynomial FreimanRuzsa conjecture over function fields. In particular, we prove that if A subset of F-p[t] satisfies |A + tA| <= K |A| then A is efficiently covered by at most K-O(1) translates of a generalised arithmetic progression of rank O(logK) and size at most K-O(1) |A|. As an application we give an optimal lower bound for the size of A + xi A where A subset of F-p((t(-1))) is a finite set and xi is an element of F-p((t(-1))) is transcendental over F-p[t].
Coven and Meyerowitz [1] formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group Z(M). In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions f , g, such that f(0) = g(0) = 1, f & lowast; g = 1(ZM) is a functional tiling, and f, g satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.
This paper studies the stability of tensor ranks under field extensions. Our main contributions are fourfold: (1) We prove that the analytic rank is stable under field extensions. (2) We establish the equivalence between the partition rank vs. analytic rank conjecture and the stability conjecture for partition rank. We also prove that they are equivalent to other two important conjectures. (3) We resolve the Adiprasito-Kazhdan-Ziegler conjecture on the stability of the slice rank of linear subspaces under field extensions. (4) As an application of (1), we show that the geometric rank is equal to the analytic rank up to a constant factor.
We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.
