Related papers: On a density conjecture about unit fractions

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

We show that for any set $A \subset \mathbb{N}$ with positive upper density and any $\ell,m \in \mathbb{N}$, there exist an infinite set $B\subset \mathbb{N}$ and some $t\in \mathbb{N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\…

Let $\{u_n\}_{n=1}^{\infty}$ be the Sylvester's sequence (sequence A000058 in the OEIS), and let $ a_1 < a_2 < \cdots $ be any other positive integer sequence satisfying $ \sum_{i=1}^\infty \frac{1}{a_i} = 1 $. In this paper, we solve a…

We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.

The author \cite{4} proved that, for every set $S$ of positive integers containing 1 (finite or infinite) there exists the density $h=h(E(S))$ of the set $E(S)$ of numbers whose prime factorizations contain exponents only from $S,$ and gave…

We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying \[\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0\] there exist a…

Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha)$ in the inequality…

We prove that if $A$ is any set of prime numbers satisfying \[ \sum_{a\in A}\frac{1}{a}=\infty, \] then $A$ must contain a $3$-term arithmetic progression. This is accomplished by combining the transference principle with a density…

For a class of arithmetic subgroups $\Gamma$ in SU(d,1) we prove that for every positive integer $n$ there exists a subgroup $\Gamma_n$ of finite index in $\Gamma$, which lifts to the $n$-fold connected cover of of SU(d,1). Consequently…

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

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…

In a recent paper, Kolountzakis and Papageorgiou ask if for every $\epsilon \in (0,1]$, there exists a set $S \subseteq \mathbb{R}$ such that $\vert S \cap I\vert \geq 1 - \epsilon$ for every interval $I \subset \mathbb{R}$ with unit…

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set…

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$,…

Let $r(k,A,n)$ denote the number of representations of $n$ as a sum of $k$ elements of a set $A \subseteq \mathbb{N}$. In 2002, Dombi conjectured that if $A$ is co-infinite, then the sequence $(r(k,A,n))_{n \geq 0}$ cannot be strictly…

We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$ having the property…

Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}~~~\text{and}~~~ R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|}\,, \] where…

Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with co\"efficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit…