Dense infinite $B_h$ sequences

TL;DR

This paper proves the existence of infinite $B_h$ sequences with specific growth rates for $h=3$ and $h=4$, extending previous work for $B_2$ sequences and providing new constructions in additive number theory.

Contribution

It establishes the existence of infinite $B_h$ sequences with precise counting functions for $h=3$ and $h=4$, extending Ruzsa's construction for $B_2$ sequences.

Findings

Existence of infinite $B_3$ sequences with specified growth rate.

Existence of infinite $B_4$ sequences with specified growth rate.

Extension of Ruzsa's $B_2$ sequence construction.

Abstract

For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\B$ with counting function $$\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$$ This result extends a construction of I. Ruzsa for $B_2$ sequences.