On arithmetic sums of Cantor-type sequences of integers

TL;DR

This paper explores the structure of sum sets of Cantor-like integer sequences, proving that for a specific binary sequence the sum of its subset sums covers all natural numbers, and extends results to other bases.

Contribution

It introduces a novel analysis of sum sets of Cantor-type sequences of integers and establishes a complete sum set for the case p=2, with structural results for p>2.

Findings

For p=2, FS(B)+FS(B)=N, covering all natural numbers.

Structural theorems for sum sets when p>2.

Analysis of subset sums of Cantor-like sequences.

Abstract

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^n\alpha\rfloor\}^\infty_{n=0}$. When $p=2$, then we prove $FS(B)+FS(B)=\N$ by analogy with the Cantor set, and some structure theorem for $p>2$