Large gaps between sums of two squarefull numbers

Abstract: A positive integer $n$ is called squarefull or powerful if in its factorization $n = p_1^{\alpha_1}\dots p_r^{\alpha_r}$ we have $\alpha_i\ge2$ for all $i$. We consider that $0$ is also a squarefull number. Thus, a number is squarefull if and only if it can be represented as $n = a^2b^3$ for some $a,b\in{\Bbb Z}_+$.

Let $W$ be the set of all sums of two squarefull numbers. Blomer (2005) proved that $$ W(x):= |W\cap[1,x]| = \frac x{(\log x)^{\alpha+o(1)}}\quad(x\to\infty),$$ where $\alpha = 1 - 2^{-1/3} = 0.20\dots$.

As suggested by Shparlinski, we study large gaps between elements of $W$. Namely, for $x > 1$ define $M(x)$ as the length of the largest subinterval of $[1,x]$ without elements of $W$. Blomer's result implies that $M(x) \ge (\log x)^{\alpha+o(1)}$ as $x\rightarrow \infty$ since the largest gap is at least as large as the average gap. We improve this estimate.

Theorem: For $x\ge3$ we have $M(x)\ge c(\log x)/(\log\log x)^2$ where $c>0$ is an absolute constant.

Joint work with Alexander Kalmynin.

combinatoricsnumber theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANY 2023)