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.
