Related papers: Arithmetic structures in random sets

This paper presents an exposition of Rio's proof of the strong law of large numbers and extends his method to random fields. In addition to considering the rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers, we go…

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds…

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset…

Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$-element subset $B$ of its vertex set. Focussing on the case that the degrees of vertices in $\mathcal{H}$…

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater…

An idea that became unavoidable to study zero entropy symbolic dynamics is that the dynamical properties of a system induce in it a combinatorial structure. An old problem addressing this intuition is finding a structure theorem for…

For many standard models of random structure, first-order logic sentences exhibit a convergence phenomenon on random inputs. The most well-known example is for random graphs with constant edge probability, where the probabilities of…

We obtain a version of the theorem of the square and a local structure result for actions of connected algebraic groups on seminormal varieties in characteristic 0, and arbitrary varieties in positive characteristics.

In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.

Let A be a subset of Z / NZ, and let R be the set of large Fourier coefficients of A. Properties of R have been studied in works of M.-C. Chang and B. Green. Our result is the following : the number of quadruples (r_1, r_2, r_3, r_4) \in…

We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected density of k-term arithmetic progressions…

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can…

Let $0\le \alpha \le \beta\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = \alpha$ and $\bar{d}(A+B) = \beta$, where $\underline{d}(A+ B)$ and…

Let $A$ be a subset of $\mathbb{Z} / N\mathbb{Z}$ and let $\mathcal{R}$ be the set of large Fourier coefficients of $A$. Properties of $\mathcal{R}$ have been studied in works of M.-- C. Chang, B. Green and the author. In the paper we…

The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search…

Previously, the author introduced quasirandom permutations, permutations of $\mathbb{Z}_n$ which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly…

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]\colon\text{deg}~ f \le N\}$ contains no pair of elements whose…

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many…

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles…