On Sums of Semibounded Cantor Sets

TL;DR

This paper explores conditions under which sums of functions of semibounded closed sets on the real line contain half-lines, with implications for spectral theory in aperiodic order.

Contribution

It establishes new criteria involving thickness and growth assumptions that guarantee the sum of such sets contains half-lines, and demonstrates the sharpness of these criteria.

Findings

Sums contain half-lines under certain thickness and growth conditions

Criteria are shown to be sharp through specific examples

Results have implications for spectral theory of aperiodic models

Abstract

Motivated by questions arising in the study of the spectral theory of models of aperiodic order, we investigate sums of functions of semibounded closed subsets of the real line. We show that under suitable thickness assumptions on the sets and growth assumptions on the functions, the sums of such sets contain half-lines. We also give examples to show our criteria are sharp in suitable regimes.