SEMINARIO DE TEORÍA DE NÚMEROS

STRONG SIDON SEQUENCES

SPEAKER: Juanjo Rué (UPC)

DATE & TIME: Tuesday, May 25th, 2021 - 12:30 !!!!!!!!!!!!!!!


ABSTRACT: A set of integers $S subset N$ 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
%
$$
big| (x+w) - (y+z) big| geq max { x^{alpha},y^{alpha},z^{alpha},w^alpha }
$$
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for every $x,y,z,w in S$ satisfying $max {x,w}
eq max {y,z}$. We obtain a new lower bound for the growth of $alpha$--strong infinite Sidon sets when $0 leq alpha < 1$. 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 R"odl.

This is a joint work with David Fabian (FU Berlin) and Christoph Spiegel (ZIB Berlin)