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Register a new userA Weighted Prekopa-Leindler inequality and sumsets with quasicubes
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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.
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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 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.
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 is known that |(h -- 1)A| $ge$ |hA| (h--1)/h , a consequence of Pl{u}nneckes inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulays 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $ge$ $theta$(x, h) |hA| (h--1)/h for some factor $theta$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $theta$(x, h) asymptotically tends to e $approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.
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 lattices. Namely, we use the finite-dimensional representation theory of sl_2 to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus-Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.
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