Related papers: Variations on the Erd\H{o}s distinct-sums problem

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…

The sum of square roots is as follows: Given $x_1,\dots,x_n \in \mathbb{Z}$ and $a_1,\dots,a_n \in \mathbb{N}$ decide whether $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$. It is a prominent open problem (Problem 33 of the Open Problems Project),…

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…

Let $a(n)$ be the number of partitions of $n$ of the form $a_1 + a_2 + \cdots + a_k$ where $a_{i + 1}$ is a proper divisor of $a_i$ for all $i < k$. Erd{\H o}s and Loxton showed that the sum of $a(n)$ over all $n \leq x$ is asymptotic to a…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear…

In 1935, Erd\H{o}s proved that the sums $f_k=\sum_n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum_p 1/(p\log…

Erd\H{o}s' unit distance problem and Erd\H{o}s' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of…

According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$ sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a discrepancy…

Solving a problem by Erd\H{o}s, we prove that every positive integer $n$ can be written as a sum $$n = b_{1} + b_{2} + \ldots + b_{r}$$ of distinct $3$-smooth integers with $1 \le b_{1} < b_{2} < \ldots < b_{r} < 6b_{1}$.

Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<\alpha|A|$ and $\alpha \ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a…

Pilz's conjecture states that for any finite set $A=\{a_1,a_2,\dots,a_k\}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least…

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.

Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in…

A sequence of positive integers $(a_1,a_2,\ldots,a_k)$ is called $\ell$-additive if $a_1+a_2+\cdots+a_k=\ell a_1$ or $\ell a_k$. In this paper, we prove that for all $k\geq3$, if $n$ is sufficiently large, then every permutation of…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

The Erd\"{o}s-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erd\"{o}s-Straus equation, state…

Erd\H{o}s posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question…