On a sumset problem for integers

TL;DR

This paper investigates lower bounds for the size of sumsets of the form A + k·A for finite integer sets, establishing new inequalities for specific values of k, especially prime powers and products of two primes.

Contribution

The paper introduces new lower bounds for sumsets involving integer sets and specific multipliers, extending previous results to prime power and product-of-two-primes cases.

Findings

Proved |A + k·A| ≥ (k+1)|A| - ⌈k(k+2)/4⌉ for prime power or product of two primes k.

Established |A + 4·A| ≥ 5|A| - 6 for |A| ≥ 5.

Provided bounds for sumsets when |A| is sufficiently large.

Abstract

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq 5|A|-6 $ for $|A|\geq 5$.