Abstract Let $\psi :\mathbb{N}\rightarrow \mathbb{R}_{+}$ be a monotonically non-increasing function, and let $\psi _{v}:\mathbb{N}\rightarrow \mathbb{R}_{+}$ be defined by $\psi _{v}(q)=1/q^{v}$. In this article, we consider self-similar sets whose iterated function systems satisfy the open set condition. For functions $\psi $ that do not decrease too rapidly, we give a conjecturally sharp upper bound on the Hausdorff dimension of the intersection of $\psi $-well approximable numbers and such self-similar sets. When $\psi =\psi _{v}$ for some $v$ greater than 1 and sufficiently close to $1$, we give a lower bound for this Hausdorff dimension, which asymptotically matches the upper bound as $v\downarrow 1$. In particular, we show that the set of very well approximable numbers has full Hausdorff dimension within self-similar sets, thus confirming a conjecture of Levesley, Salp, and Velani.
Abstract In this note, we prove that $K_{2i} (\mathbb{Z}/p^{n}) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus, and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_{1}$ in $\pi _{*} K(\mathbb{Z}/p^{n})/p$ is equal to $\frac{p^{n}-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy, and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_{K} /\varpi ^{n}$ constructed by Antieau, Nikolaus, and the first author.
Abstract A $k$-configuration is a collection of $k$ distinct integers $x_{1},\ldots ,x_{k}$ together with their pairwise arithmetic means $\frac{x_{i}+x_{j}}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini, and Kelman on binary systems of linear forms and of Kelley and Meka on Roth’s theorem on arithmetic progressions, we show that, for $N \geq \exp ((k\log (2/\alpha ))^{O(1)})$, any subset $A \subseteq [N]$ of density at least $\alpha $ contains a $k$-configuration. This improves on the previously best known bound $N \geq \exp ((2/\alpha )^{O(k^{2})})$, due to Shao. As a consequence, it follows that any finite non-empty set $A \subseteq \mathbb{Z}$ contains a subset $B \subseteq A$ of size at least $(\log |A|)^{1+\Omega (1)}$ such that $b_{1}+b_{2} \not \in A$ for any distinct $b_{1},b_{2} \in B$. This provides a new proof of a lower bound for the Erdős–Moser sum-free set problem of the same shape as the best known bound, established by Sanders.
Abstract We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform shown by Biane, Bougerol, and O’Connell. This defines a group action of the infinite symmetric group on sequences of vectors in $\mathbb R^{\mathbb Z_{N}}$. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. Combined with a new inhomogeneous Burke property for the periodic Pitman transform, we prove a multi-path invariance result for the periodic inverse-gamma polymer under permutations of the column parameters. In the limit to the full-line case, we obtain a multi-path extension of a recent invariance result of Bates, Emrah, Martin, Seppäläinen, and the fifth author, in both positive and zero-temperature.
Abstract Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_{1}+b_{2}\colon b_{1},b_{2}\in B\}$ or $\{b_{1}+b_{2}\colon b_{1},b_{2}\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.
