Related papers: Effective bounds for Roth's theorem with shifted s…

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.

We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension.…

For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$\Phi_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove…

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*}…

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of…

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 study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially…

We prove that a point $x$ is normal with respect to an ergodic, number-theoretic transformation $T$ if and only if $x$ is normal with respect to $T^n$ for any $n\ge 1$. This corrects an erroneous proof of Schweiger. Then, using some…

A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we…

Let $m\geq 2$ be a positive integer. Given a set $E(\omega )\subseteq \mathbb{N}$ we define $r_{N}^{(m)}(\omega )$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\textit{ and }$ differences of $m$…

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},\mu)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_{\mu}(X)$ we have that the…

We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c.

Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. Assume that $h(\cdot)$ is a positive non-decreasing…

Let $\alpha_1, \cdots, \alpha_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\alpha_i s||_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $i$ and some fixed $\delta>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is…

We establish a stability result for the Shannon-McMillan-Breiman theorem on the one-sided finite shift space. For any shift-invariant probability measure P and any data-dependent parsing whose number of blocks is sublinear in N almost…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a pair of conjugate…

Erd\H os introduced the quantity $S=T\sum^T_{i=1}X_i$, where $X_1,\dots, X_T$ are arithmetic progressions, and cover the square numbers up to $N$. He conjectured that $S$ is close to $N$, i.e. the square numbers cannot be covered…

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$