On some direct and inverse results concerning sums of dilates
Volume 188 / 2019
Acta Arithmetica 188 (2019), 101-109
MSC: Primary 11P70; Secondary 11B13, 11B75.
DOI: 10.4064/aa170623-25-6
Published online: 18 February 2019
Abstract
Let $A$ and $B$ be two nonempty finite sets of integers and let $r$ be a positive integer. Define $A+r\cdot B:=\{a+rb:a\in A,\, b\in B \}$. In case $A=B$, Freiman et al. proved that $|A+r\cdot A|\geq 4|A|-4$ for $r\geq 3$. For $r=2$, they obtained an extended inverse result which states that if $|A|\geq 3$ and $|A+2\cdot A| \lt 4|A|-4$, then $A$ is a subset of an arithmetic progression of length at most $2|A|-3$. We present a new, self-contained proof of the direct result, $|A+r\cdot A|\geq 4|A|-4$ for $r\geq 3$. We also generalize the above extended inverse result to sums $A+2\cdot B$ for two sets $A$ and $B$.
