An exploration of Nathanson's $g$-adic representations of integers

TL;DR

This paper investigates Nathanson's $g$-adic representations to connect Cayley graph properties of integers with additive number theory, providing explicit formulas and bounds related to powers of integers and primes, and linking to Goldbach's conjecture.

Contribution

It introduces explicit formulas for minimal integers of a given length in $g$-adic representations and establishes bounds on Cayley graph diameters related to additive problems.

Findings

Explicit formulas for smallest integers of a given length in $g$-adic representations.

Bounds on Cayley graph diameters related to Goldbach's conjecture.

Connections between metric properties of Cayley graphs and additive number theory.

Abstract

We use Nathanson's $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number theory. If $S$ consists of all powers of a fixed integer $g$, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of $g$. We also consider $S$ to be the set of all powers of all primes and bound the diameter of Cayley graph by relating it to Goldbach's conjecture.