On a theorem of Sárközy for difference sets and shifted primes

Difference sets and the primes

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.

Tiling sets and spectral sets over finite fields

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to $2$ it has recently been proven that the Fuglede conjecture holds (see \cite{IMP15}). In t…

Almost arithmetic progressions in the primes and other large sets

A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note I provide a straightforward argument demonstrating that the primes get arbitrarily close to arbitrarily long arithmetic progressions. The argument also applies to `large sets' in the sense of Erdős-Turán. The proof is short, completely self-contained, and aims to give a heuristic explanation of why the primes, and other large sets, possess arithmetic structure.

Performances piecewise defined functions in analytic form, prime-counting function, $ξ$ sets

The article discusses the representation of discrete functions defined in an analytic form without the use of approximations, namely the Heaviside function, identity function, the Dirac delta function and the prime-counting function. Also in the article introduced and considered a new type of sets ($ξ$ set) by analogy taken from finding the sum of a number of grants and other contradictions in mathematics and physics. With the help of $ξ$ sets interpreted Russell's paradox in the system of axioms of naive set theory.

Peres--Schlag 非空内部问题与乘积集的移位乘积变体

Peres--Schlag's nonempty-interior problem and a shifted-product variant for product sets

研究 Peres--Schlag 非空内部问题在有限域和欧几里得空间中的乘积集变体，证明当集合大小或维数超过阈值时，线性组合或移位乘积覆盖整个空间或包含开区间。

$\ell^p(\mathbb{Z})$ -- boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems

We establish the first pointwise ergodic theorems along thin sets of prime numbers; a set with zero density with respect to the primes. For instance we will be able to achieve this with the Piatetski-Shapiro primes. Our methods will be robust enough to solve the ternary Goldbach problem for some thin sets of primes.