Random Sidon Sequences
نویسندگان
چکیده
A subset A of the set [n] = f1; 2; : : : ; ng, jAj = k, is said to form a Sidon (or Bh) sequence, h 2, if each of the sums a1 + a2 + : : : + ah; a1 a2 : : : ah; ai 2 A, are distinct. We investigate threshold phenomena for the Sidon property, showing that if An is a random subset of [n], then the probability that An is a Bh sequence tends to unity as n ! 1 if kn = jAnj n1=2h, and that P(An is Sidon) ! 0 provided that kn n1=2h. The main tool employed is the Janson exponential inequality. The validity of the Sidon property at the threshold is studied as well; we prove, using the Stein{ Chen method of Poisson approximation, that P(An is Sidon) ! expf g (n ! 1) if kn n1=2h ( 2 R+), where is a constant that depends in a well-speci ed way on . Multivariate generalizations are presented. 3
منابع مشابه
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