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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.
In this paper, we prove a Prekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prekopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.
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 availa
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.
Let A be a finite subset of an abelian group (G, +). Let h $ge$ 2 be an integer. If |A| $ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $times$ $times$ $times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean la