Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa166-1-5,
author = {Kathryn E. Hare and Shuntaro Yamagishi},
title = {A generalization of a theorem of Erd\H os-R\'enyi to m-fold sums and differences},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {55-67},
zbl = {1306.11060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa166-1-5}
}
Kathryn E. Hare; Shuntaro Yamagishi. A generalization of a theorem of Erdős-Rényi to m-fold sums and differences. Acta Arithmetica, Tome 166 (2014) pp. 55-67. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa166-1-5/