Title

Random Sidon Sequences

Document Type

Article

Publication Date

3-1-1999

Abstract

A subsetAof the set [n]={1,2,...,n}, A=k, is said to form aSidon(orBh) sequence,h≥2, if each of the sumsa1+a2+...+ah,a1≤a 2≤...≤ah;ai∈A, are distinct. We investigate threshold phenomena for the Sidon property, showing that ifAnis a random subset of [n], then the probability thatAnis aBhsequence tends to unity asn→∞ ifkn=An≪n1/2h, and thatP(Anis Sidon)→0 provided thatkn≫n1/2h. The main tool employed is the Janson exponential inequality. The validity of the Sidon propertyatthe threshold is studied as well. We prove, using the Stein-Chen method of Poisson approximation, thatP(Anis Sidon) →exp{-λ} (n→∞) ifkn~Lambda;·n1/2h(Λ∈R+), whereλis a constant that depends in a well-specified way onΛ. Multivariate generalizations are presented. © 1999 Academic Press.

Publication Title

Journal of Number Theory

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