On possible sums from multiset of powers of $d$

TL;DR

This paper characterizes the structure of sums from multisets of powers of an integer and provides criteria for when two such multisets generate the same sum set.

Contribution

It introduces a detailed description of the sum set structure and establishes a criterion for equality of sum sets from different multisets of powers.

Findings

Explicit structure of sum sets from multisets of powers of d

Criterion for equality of sum sets from different multisets

Theoretical framework for analyzing sums of powers

Abstract

Let $A$ be a finite multiset of powers of a positive integer $d>1$. We describe the structure of the set $\mathrm{span}(A)$ of all sums of submultisets of $A$, and in particular give a criterion of $\mathrm{span}(A)=\mathrm{span}(B)$ for two multisets $A,B$.