Representing integers as linear combinations of powers

TL;DR

This paper investigates how integers can be represented as linear combinations of powers of fixed integers, extending previous results on sums and differences of powers of 2 and 3.

Contribution

It provides new theoretical results on representing integers as linear combinations of powers of fixed integers, generalizing earlier work on sums of powers of 2 and 3.

Findings

Proved new bounds for integer representations as linear combinations of powers.

Extended results to more general sets of fixed integers.

Enhanced understanding of arithmetic diameters of such sets.

Abstract

At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by sums or differences of powers of 2 and 3. In this note we prove some results on this problem and the more general problem about the representation by linear combinations of powers of some fixed integers.