A set of integers S subset of N is an alpha-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on alpha, more specifically if vertical bar(x + w) - (y + z)vertical bar &...
详细信息
A set of integers S subset of N is an alpha-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on alpha, more specifically if vertical bar(x + w) - (y + z)vertical bar >= max{x(alpha), y(alpha), z(alpha), w(alpha)} for every x, y, z, w is an element of S satisfying max{x, w} not equal max{y, z}. We obtain a new lower bound for the growth of alpha-strong infinite Sidon sets when 0 <= alpha < 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for alpha-strong infinite B-h sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or B-h set contained in a random infinite subset of N. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rodl. (C) 2021 Elsevier Inc. All rights reserved.
暂无评论