Related papers: Large sets avoiding linear patterns

We demonstrate a compactness result holding broadly across supervised learning with a general class of loss functions: Any hypothesis class $H$ is learnable with transductive sample complexity $m$ precisely when all of its finite…

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products,…

Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $\Gamma\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$,…

We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets…

The main aim of this work is to show, in the absence of the Axiom of Choice, fundamental results on $\mathbf{E}$-compact extensions of $\mathbf{E}$-completely regular spaces, in particular, on Hewitt realcompactifications and Banaschewski…

A classical result in combinatorial number theory states that the largest subset of $[n]$ avoiding a solution to the equation $x+y=z$ is of size $\lceil n/2 \rceil$. For all integers $k>m$, we prove multicolored extensions of this result…

We study some variants of the Erd\H{o}s similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset…

While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the…

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial…

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subset of the line $E$ and an arbitrary superlinearity, there exists a smooth, strictly convex…

We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\'edi's theorem in the integers…

We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it…

We define the infinitesimal and geometric orders of an effective Klein geometry G/H. Using these concepts, we prove i) For any integer m>1, there exists an effective Klein geometry G/H of infinitesimal order m such that G/H is a projective…

A matching of the set $[2n]=\{ 1,2,\ldots ,2n\}$ is a partition of $[2n]$ into blocks with two elements, i.e. a graph on $[2n]$ such that every vertex has degree one. Given two matchings $\sigma$ and $\tau$ , we say that $\sigma$ is a…

We give an example of a countable theory T such that for every cardinal lambda >= aleph_2 there is a fully indiscernible set A of power lambda such that the principal types are dense over A, yet there is no atomic model of T over A. In…

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if…

We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard…

We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set $K\subset\mathbb{R}^d$ with Assouad dimension arbitrarily close to $d$ such that every microset of…

We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such…