A paucity problem associated with a shifted integer analogue of the divisor function

TL;DR

This paper investigates the uniqueness of representing elements in the ring generated by a shifted irrational integer, revealing that almost all such elements are essentially uniquely expressed as a product of shifted integers.

Contribution

It introduces a new perspective on the representation of elements in the shifted integer ring, highlighting a scarcity problem related to the divisor function analogue.

Findings

Almost all elements in the ring are uniquely represented as products of shifted integers.

The study extends understanding of divisor function analogues in algebraic number theory.

Highlights a scarcity problem in the representation of elements in shifted integer rings.

Abstract

Suppose that $\theta$ is irrational. Then almost all elements $\nu\in {\mathbb Z}[\theta]$ that may be written as a $k$-fold product of the shifted integers $n+\theta$ $(n\in {\mathbb N})$ are thus represented essentially uniquely.