The third positive element in the greedy $B_h$-set
Melvyn B. Nathanson
TL;DR
The paper investigates greedy $B_h$-sets, focusing on the third element $a_3(h)$. It leverages the key inequality $a_{k+1}(h)\le h\,a_k(h)+1$ and a combinatorial nonexistence argument to pin down $a_3(h)$. The main result is that $a_3(h)=h^2+h+1$ for all $h\ge1$, and it establishes the general bound $a_k(h)\le \sum_{i=0}^{k-1} h^i$, with further discussion of the early growth, including that $a_4(h)$ is a quasi-polynomial with explicit odd/even forms, later confirmed in subsequent work. These results sharpen the understanding of the early structure of greedy $B_h$-sets and contribute to the broader study of Sidon and $B_h$-sets in additive number theory.
Abstract
For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_j} \in A$ for all $j = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$ set. One has $a_1(h) = 1$ and $a_2(h) = h+1$ for all $h$. Elementary proofs are given that $a_3(h) = h^2+h+1$ for all $h \geq 1$ and that $a_k(h) \leq \sum_{i=0}^{k-1} h^i$ for all $h \geq 1$ and $k \geq 1$.