No Arabic abstract
Given a finite set of integers $A$, its sumset is $A+A:= {a_i+a_j mid a_i,a_jin A}$. We examine $|A+A|$ as a random variable, where $Asubset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p in (0,1)$. Recently, Martin and OBryant studied the case in which $p=1/2$ and found a closed form for $mathbb{E}[|A+A|]$. Lazarev, Miller, and OBryant extended the result to find a numerical estimate for $text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n,;,p}(k) := mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $mathbb{E}[|A+A|]$ and $text{Var}(|A+A|)$ for all $pin (0,1)$ and establish good bounds for $mathbb{E}[|A+A|]$ and $m_{n,;,p}(k)$. We continue to investigate $m_{n,;,p}(k)$ by studying $m_p(k) = lim_{ntoinfty}m_{n,;,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and OBryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)<m_{1/2}(8)$. This distribution is not unimodal, and is said to have a divot at 7. We report results investigating this divot as $p$ varies, and through both theoretical and numerical analysis, prove that for $pgeq 0.68$ there is a divot at $1$; that is, $m_{p}(0)>m_{p}(1)<m_{p}(2)$. Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and OBryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $mathbb{P}(iin B mid iin A) = p_1$ and $mathbb{P}(iin B mid i otin A) = p_2$. We provide some preliminary results using the extension of this framework.
In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q rightarrow infty$ using Steinhaus random multiplicative functions, stating properties of this random process. This is motivated by Kowalski and Sawins work on Kloosterman paths.
We obtain an upper bound for the number of pairs $ (a,b) in {Atimes B} $ such that $ a+b $ is a prime number, where $ A, B subseteq {1,...,N }$ with $|A||B| , gg frac{N^2}{(log {N})^2}$, $, N geq 1$ an integer. This improves on a bound given by Balog, Rivat and Sarkozy.
We consider very general random integers and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
Let $d$ be a positive integer and $U subset mathbb{Z}^d$ finite. We study $$beta(U) : = inf_{substack{A , B eq emptyset text{finite}}} frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$beta(U) = |U|,$$ whenever $U$ is a subset of ${0,1}^d$. Our methods parallel those used for the Prekopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.