Random Sidon Sequences
Document Type
Article
Publication Date
3-1-1999
Department
Department of Mathematical Sciences
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.
Publication Title
Journal of Number Theory
Recommended Citation
Godbole, A.,
Janson, S.,
Locantore, N.,
&
Rapoport, R.
(1999).
Random Sidon Sequences.
Journal of Number Theory,
75(1), 7-22.
http://doi.org/10.1006/jnth.1998.2325
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/3985