Bounds for Greedy $B_h$-sets

Abstract

A set $A$ of nonnegative integers is called a $B_h$-set if every solution to $a_1+\cdots +a_h=b_1+\cdots +b_h$, where $a_i,b_i\in A$, has $\{a_1,\dots ,a_h\}=\{b_1,\dots ,b_h\}$ (as multisets). Let ${\hat{I}}_k^3(h)$ be the $k$-th positive element of the greedy $B_h$-set. We give a nontrivial lower bound on ${\hat{I}}_5^3(h)$, and a nontrivial upper bound on ${\hat{I}}_k^3(h)$ for $k\ge 5$. Specifically, $\frac{1}{8}{h}^4+\frac{1}{2}{h}^3\le {\hat{I}}_5^3(h)\le 0.467214h^4+O(h^3)$, although we conjecture that ${\hat{I}}_5^3(h)=\frac{1}{3}{h}^4+O(h^3)$. We show that ${\hat{I}}_k^3(h)\ge \frac{1}{k!}{h}^{k-1}+O(h^{k-2})$ for $k\ge 1$ and ${\hat{I}}_k^3(h)\le \hat{I}{\pm }_k{h}^{k-1}+O(h^{k-2})$, where $\hat{I}{\pm }_6:=0.382978$, $\hat{I}{\pm }_7:=0.269877$, and for $k\ge 7$, $\hat{I}{\pm }_{k+1}:=\frac{1}{2^{k}k!}{\sum }_{j=0}^{k-1}\left(\genfrac{}{}{0ex}{}{k-1}{j}\right)\left(\genfrac{}{}{0ex}{}{k}{j}\right)2^j$. This work begins with a thorough introduction and concludes with a section of open problems.

19 pages, including appendix with proof details omitted from journal submission (this version has an improved introduction)