On additive and multiplicative decompositions of sets of integers with restricted prime factors, II. (Smooth numbers and generalizations.)

TL;DR

This paper extends the study of additive and multiplicative decompositions of smooth numbers and their shifted sets, focusing on large functions y and exploring m-decomposability of k-term sumsets.

Contribution

It advances understanding of decomposability properties of smooth numbers for large y and generalizes to m-decomposability of sumsets with arbitrary k.

Findings

Analyzed additive decomposability of y-smooth numbers for large y

Investigated multiplicative decomposability of shifted smooth number sets

Explored m-decomposability of k-term sumsets

Abstract

In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.