Sidon sets in N{double-struck}~d

摘要

We study finite and infinite Sidon sets in N{double-struck}~d. The additive energy of two sets is used to obtain new upper bounds for the cardinalities of finite Sidon subsets of some sets as well as to provide short proofs of already known results. We also disprove a conjecture of Lindstr?m on the largest cardinality of a Sidon set in [1,N]×[1,N] and relate it to a known conjecture of Vinogradov concerning the size of the smallest quadratic residue modulo a prime p.For infinite Sidon sets A?N{double-struck}~d, we prove that lim inf_(n→∞){pipe}A∩[1,n]~d{pipe}/√n~d/logn>0. Finally, we show how to map infinite Sidon sets in N{double-struck}~d to N{double-struck}~d′ in an effective way. As an application, we find an explicit Sidon set of positive integers A such that {pipe}A∩[1,n]{pipe}≥n~(1/3+o(1)).