Related papers: Unbounded logarithmic limsup in Erd\H{o}s problem …

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…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which…

Itsykson and Sokolov [IS14] identified resolution over parities, denoted by $\text{Res}(\oplus)$, as a natural and simple fragment of $\text{AC}^0[2]$-Frege for which no super-polynomial lower bounds on size of proofs are known. Building on…

One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot…

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…

The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case…

The Erd\H{o}s-Ginzburg-Ziv theorem states that every sequence of 2n - 1 integers contains a subsequence of length n whose sum is divisible by n. Choi, Kang, and Lim gave a simple deterministic O(n log n) algorithm for finding such a…

For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the…

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed $m$ there are at most $\mathcal{O}_{\epsilon}(n^{3/5+\epsilon})$ solutions of…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

Fix an integer $k\ge 3$. Call a set $A\subseteq [N]$ LCM-$k$-free if it does not contain distinct $a_1,\dots,a_k$ such that $\mathrm{lcm}(a_i,a_j)$ is the same for all $1\le i<j\le k$. Define $$ f_k(N):=\max\left\{\sum_{a\in A}\frac1a:…

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\leq x$ such that a non-empty set of irreducible polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this set can…

Typically, one expects that there are around x\prod_{p\not\in P, p <= x} (1-1/p) integers up to x, all of whose prime factors come from the set P. Of course for some choices of P one may get rather more integers, and for some choices of P…

Given $d,s \in \mathbb{N}$, a finite set $A \subseteq \mathbb{Z}$ and polynomials $\varphi_1, \dots, \varphi_{s} \in \mathbb{Z}[x]$ such that $1 \leq deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{(s)}| +…

We give a new proof of F\"urer's bound for the cost of multiplying n-bit integers in the bit complexity model. Unlike F\"urer, our method does not require constructing special coefficient rings with "fast" roots of unity. Moreover, we prove…

We prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any…

Since their introduction by Erd\H{o}s in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding…

The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of…

Let $k \geq 3$ and let $L_i(n) = A_in + B_i$ be some linear forms such that $A_i$ and $B_i$ are integers. Define ${\mathcal{P}(n) = \prod_{i=1}^k L_i(n)}$. For each $k$ it is known that $\Omega (\mathcal{P} (n) ) \leq \rho_k$ infinitely…