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

Greg Bell; Austin Lawson; Neil Pritchard; Dan Yasaki

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

Geometric Topology 2019-01-21 v2

Authors: Greg Bell , Austin Lawson , Neil Pritchard , Dan Yasaki

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.

Keywords

g-adic representation cayley graph additive number theory infinite generating set goldbach conjecture integer lengths

Cite

@article{arxiv.1711.00809,
  title  = {An exploration of Nathanson's $g$-adic representations of integers},
  author = {Greg Bell and Austin Lawson and Neil Pritchard and Dan Yasaki},
  journal= {arXiv preprint arXiv:1711.00809},
  year   = {2019}
}

Comments

10 pages, rewritten to replace "On locally infinite Cayley graphs of Z."

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

Abstract

Keywords

Cite

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