Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn analytic approach to cardinalities of sumsets
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$ can be made isomorphic by changing o(n^2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdos-Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fraction
In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random walk to provide such estimates. We also include here an upper bound, derived by elementary methods, on the number of partial Hadamard.
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|.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
