Related papers: More pointsets with many rich lines
In this article we describe all possible infinite linear configurations that can be found in a shift of any set of positive upper Banach density. This simultaneously generalizes Szemer\'edi's theorem on arithmetic progressions and the…
We apply techniques from symplectic geometry to extend and give a new proof of the complex convexity theorem of Gindikin-Kroetz.
This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader…
Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we…
In this note we improve an algorithm from a recent paper by Bauer and Bennett for computing a function of Erd\"os that measures the minimal gap size $f(k)$ in the sequence of integers at least one of whose prime factors exceeds $k$. This…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We…
We survey what is known about Fekete points/optimal designs for a simplex in $\R^d.$ Several new results are included. The notion of Fej\'er exponenet for a set of interpolation points is introduced.
Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a…
We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemer\'edi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we…
We introduce a new approach for the study of the Problem of Iterates using the theory on general ultradifferentiable structures developed in the last years. Our framework generalizes many of the previous settings including the Gevrey case…
We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renner's proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all…
Building upon the theory of graph limits and the Aldous-Hoover representation and inspired by Panchenko's work on asymptotic Gibbs measures (Annals of Probability 2013), we construct continuous embeddings of discrete probability…
A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.
A major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemer\'edi's theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs…
We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the…
A celebrated result of Gowers states that for every \epsilon > 0 there is a graph G so that every \epsilon-regular partition of G (in the sense of Szemeredi's regularity lemma) has order given by a tower of exponents of height polynomial in…
The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. In addition, the second…
A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ that satisfy a Boolean combination of a…
We give an elementary and constructive proof for a theorem of de Smit et Lenstra. Note: In version 1, was missing the proof that "completely secant" implies "1-secant"