On a multiplicative hybrid problem over almost-primes

TL;DR

This paper proves the existence of almost-prime products close to a specific quadratic form within certain subsets of integers, advancing understanding of multiplicative structures over almost-primes.

Contribution

It establishes a new result connecting almost-primes and quadratic forms, extending previous work on multiplicative problems over almost-primes.

Findings

Existence of integers a, b with a in A, b in B satisfying the almost-prime quadratic relation.

Bounds on the approximation error involving almost-primes.

Extension of multiplicative problems to subsets involving almost-primes.

Abstract

Let N be a large enough natural number, A and B be subsets of {N+1, ... , 2N}. In this paper, we prove that there exists integers a, b with a belongs to A, b belongs to B such that ab=P_k^2 + O(P_k^{1-c}), where 0<c<1/2 and P_k denotes an almost-prime with at most k prime factors, counted with multiplicity.