For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq \mathbb{Z}_m$ with $1\le R_A(n)\le r$ for all $n\in \mathbb{Z}_m$. In 2008, Chen proved that $R_{m}\le 288$ for all positive integers $m$. Recently the authors proved that $R_m\ge 6$ for all integers $m\ge 36$. In this paper, we prove that if $A\subseteq \mathbb{Z}_m$ satisfies $R_A(n)\le 5$ for all $n\in \mathbb{Z}_m$, then $|\{g:g\in \mathbb{Z}_m, R_A(g)=0\}|\ge \frac{1}{4}m-\sqrt{5m}$. This improves a recent result of Li and Chen. We also give upper bounds of $|\{g:g\in \mathbb{Z}_m, R_A(g)=i\}|$ for $i=2,4$.
Let $\mathcal{P}$ denote the set of all primes. In 1950, P. Erdős conjectured that if $c$ is an arbitrarily given constant, $x$ is sufficiently large and $a_1,\dots , a_t$ are positive integers with $a_1<a_2<\cdot\cdot\cdot<a_t\leqslant x$ and $t>\log x$, then there exists an integer $n$ so that the number of solutions of $n=p+a_i$ $(p\in \mathcal{P}, 1\le i\le t)$ is greater than $c$. In this note, we confirm this old conjecture of Erdős.
The Erdös--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 $n\geq 2$. In this short note we find explicity the solutions of the famous conjecture for the case $n\equiv13 \textrm{ mod } 24.$
The Erdös-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 Erdös-Straus equation, state several methods for obtaining well-defined equations, and some features of well-defined equations. Using well-defined equations, we explain the inefficiency of the solutions that did not lead to a complete proof of the problem. Finally, we prove that this conjecture cannot be completely proven by the methods expressed in various articles. For this reason, we express con…
This paper is a preliminary expository paper that outlines the relationship between solutions to the Erdős-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses Bézout Coefficients to find a relationship between the solutions to the conjecture and roots of a second degree polynomial.
Erdős proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that $\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$, where $\mathcal{P}_j$ is the set of integers with $j$ prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field $\mathbb{F}_q[x]$, investigating the sum $\mathcal{F}…
A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.
Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
Electro-Gravitational Space Propulsion Laboratory (EGSPL)（电引力空间推进实验室）；Maulana Azad National Institute of Technology Bhopal（莫拉纳·阿扎德国家技术学院博帕尔分校）；National Taiwan University（国立台湾大学）
A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.
Erdős and Turán proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erdős-Turán inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy…
Trigonometric approximation and a general form of the Erdős Turán inequality
There exists a positive function $ψ(t)$ {on} $t\geq0$ {, with fast decay at infinity, such that for every measurable set } $Ω$ {in the Euclidean space and} $R>0$ {, there exist entire functions} $A(x) $ {and} $B(x) $ {of exponential type} $R$ {, satisfying\} $A(x)\leq χ_Ω(x)\leq B(x)$ {and} $| B(x)-A(x)| \leqslantψ(R\operatorname*{dist}(x,\partialΩ)) $ . This leads to Erdős Turán estimates for discrepancy of point set distributions in the multi dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.
