Related papers: Sets of integers that do not contain long arithmet…

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $\lambda > 2$…

Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields…

We demonstrate that recent advances in reinforcement learning (RL) combined with simple architectural changes significantly improves generalization on the ProcGen benchmark. These changes are frame stacking, replacing 2D convolutional…

We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of…

The Korkine--Zolotareff (KZ) reduction, and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr's constant were defined by Schnorr in 1987. The KZ constant can be…

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods…

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists.…

In this paper, we study bounds on the minimum length of $(k,n,d)$-superimposed codes introduced by Agarwal et al. [1], in the context of Non-Adaptive Group Testing algorithms with runlength constraints. A $(k,n,d)$-superimposed code of…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

We develop a novel method of constructing confidence bands for nonparametric regression functions under shape constraints. This method can be implemented via a linear programming, and it is thus computationally appealing. We illustrate a…

An arithmetic progression is a sequence of integers in which the difference between any two consecutive elements is the same. We investigate the parameterized complexity of two problems related to arithmetic progressions, called Cover by…

We propose a family of quantum algorithms for estimating Gowers uniformity norms $ U^k $ over finite abelian groups and demonstrate their applications to testing polynomial structure and counting arithmetic progressions. Building on recent…

This is the text accompanying my Bourbaki seminar on the work of Bloom and Sisask, Croot, Lev, and Pach, and Ellenberg and Gijswijt.

Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the model parameters. Here, we introduce an equivalent formulation of the objective function of KRR, which opens up…

Arithmetic complexity is considered simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic…

In recent years, quantitative complexity over semirings has been intensively investigated. In this context, Eiter and Kiesel (Semiring Reasoning Frameworks in AI and Their Computational Complexity, J. Artif. Intell. Res., 2023) introduced…

Let A be a subset of $\F_p^n$, the $n$-dimensional linear space over the prime field $\F_p$ of size at least $\de N$ $(N=p^n)$, and let $S_v=P^{-1}(v)$ be the level set of a homogeneous polynomial map $P:\F_p^n\to\F_p^R$ of degree $d$, and…

We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no…

Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j,…