Powered numbers in short intervals II

TL;DR

The paper studies the distribution of $k$-powered numbers in short intervals, unconditionally and under the $abc$-conjecture. It develops three main results: (i) an unconditional bound for the count when the powerful part is $y^{1-\delta}$-smooth, (ii) a bound focusing on the smooth squarefree part via a refined short-interval sieve bound and a bound on large smooth divisors, and (iii) a conditional bound (assuming $abc$) for $k>5/4$ leveraging a polynomial identity to force a contradiction with large $k$-powered APs, together with a recent density bound for AP-free sets. The methods combine sieve techniques, a polynomial identity, and recent progress on densities of $k$-term arithmetic progressions, yielding improved exponents and extending the analysis to broader notions of powerfulness. The results illuminate how smoothness constraints interact with additive-structure restrictions to bound rare, highly divisible integers in short ranges, with conditional improvements under the $abc$-conjecture.

Abstract

In this article, we derive better results concerning powered numbers in short intervals, both unconditionally and conditionally on the $abc$-conjecture. We make use of sieve method, a polynomial identity, and a recent breakthrough result on density of sets with no $k$-term arithmetic progression. In the process, we study integers over short intervals that have with a big smooth divisor.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• proof : Proof of Theorem \ref{['thm1']}
• Proposition 1
• proof
• proof : Proof of Theorem \ref{['thm2']}
• proof : Proof of Theorem \ref{['thm3']}