Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPrimes in Sumsets
70
0
0.0
(
0
)
Authors
Kummari Mallesham
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
It is proven that there are infinitely prime pairs whose difference is no greater than 20.
We use Maynards methods to show that there are bounded gaps between primes in the sequence ${lfloor nalpharfloor}$, where $alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $lfloor f(q)rfloor$ with $q$ a positive integer.
We give a short, self-contained proof of two key results from a paper of four of the authors. The first is a kind of weighted discrete Prekopa-Leindler inequality. This is then applied to show that if $A, B subseteq mathbb{Z}^d$ are finite sets and $U$ is a subset of a quasicube then $|A + B + U| geq |A|^{1/2} |B|^{1/2} |U|$. This result is a key ingredient in forthcoming work of the fifth author and Palvolgyi on the sum-product phenomenon.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
