Bounds for Greedy B_h-sets

2023/12/18 by O'Bryant, Kevin
#11B13 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.10910

Abstract

A set A of nonnegative integers is called a B_h-set if every solution to a₁+…+a_h = b₁+…+b_h, where a_i,b_i ∈ A, has \a₁,…,a_h\=\b₁,…,b_h\ (as multisets). Let γ_k(h) be the k-th positive element of the greedy B_h-set. We give a nontrivial lower bound on γ₅(h), and a nontrivial upper bound on γ_k(h) for k≥ 5. Specifically, \frac 18 h⁴ +\frac12 h³ ≤ γ₅(h) ≤ 0.467214 h⁴+O(h³), although we conjecture that γ₅(h)=\frac13 h⁴ +O(h³). We show that γ_k(h) ≥ (1)/(k!) h^k-1 + O(h^k-2) for k≥ 1 and γ_k(h) ≤ α_k h^k-1+O(h^k-2), where α₆ := 0.382978, α₇ := 0.269877, and for k≥ 7, α_k+1 := (1)/(2^k k!) ∑_j=0^k-1 \binomk-1j\binom kj 2^j. This work begins with a thorough introduction and concludes with a section of open problems.

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