Results on Sidon and B h Sequences

Results on Sidon and Bh Sequences Sangjune Lee A set A of non-negative integers is a Sidon set if all the sums a1 + a2, with a1 ≤ a2 and a1, a2 ∈ A, are distinct. In this dissertation, we deal with results on the number of Sidon sets in [n] = {0, 1, · · · , n − 1} and the maximum size of Sidon sets in sparse random subsets of [n] or N (the set of natural numbers). We also consider a natural generalization of Sidon sets called Bh-sets with h ≥ 2. A set A of non-negative integers is called a Bh-set if all the sums a1 + a2 + · · · + ah, with a1 ≤ a2 ≤ · · · ≤ ah and ai ∈ A, are distinct. The first question in this dissertation was suggested by Cameron–Erdős in 1990. They proposed the problem of estimating the number of Sidon sets contained in [n]. We obtain an upper bound 2c √ n on the number of Sidon sets which is sharp up to a constant factor in the exponent when compared to the previous lower bound 2(1+o(1)) √ n. Next, we investigate the maximum size of Sidon sets contained in sparse random sets R ⊂ [n]. Let R = [n]m be a uniformly chosen, random melement subset of [n]. Let F ([n]m) = max{|S| : S ⊂ [n]m is Sidon}. Fix a constant 0 ≤ a ≤ 1 and suppose m = (1 + o(1))na. We show that there is a constant b = b(a) for which F ([n]m) = n b+o(1) (1) almost surely and we obtain what b = b(a) is. Surprisingly, between two points a = 1/3 and a = 2/3, the function b = b(a) is constant. Next, we deal with infinite Sidon sets in sparse random subsets of N. Fix 0 < δ ≤ 1, and letR = Rδ be the set obtained by choosing each element i ⊂ N independently with probability i−1+δ. We show that for every 0 < δ ≤ 2/3 there exists a constant c = c(δ) such that a random set R satisfies the following with probability 1: • Every Sidon set S ⊂ R satisfies that |S ∩ [n]| ≤ nc+o(1) for every sufficiently large n. • There exists a large Sidon set S ⊂ R such that |S ∩ [n]| ≥ nc+o(1) for every sufficiently large n. Results on Sidon and Bh Sequences