A note on the multiplicative structure of an additively shifted product set, AA+1

TL;DR

This paper investigates the multiplicative structure of sets formed by multiplying a large finite set of real numbers and then adding one, showing such sets cannot be mostly contained within any generalized geometric progression, with analogous results in finite fields.

Contribution

It demonstrates that the shifted product set AA+1 cannot be predominantly contained within any generalized geometric progression, revealing new structural limitations.

Findings

AA+1 set has large parts outside any generalized geometric progression

Results hold for both real numbers and finite fields

Provides structural insights into shifted product sets

Abstract

We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized geometric progression of comparable length. We prove an analogous result in finite fields as well.