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On sets of integers which contain no three terms in geometric progression


Author: Nathan McNew
Journal: Math. Comp. 84 (2015), 2893-2910
MSC (2010): Primary 11B05, 11B75, 11Y60, 05D10
DOI: https://doi.org/10.1090/mcom/2979
Published electronically: May 14, 2015
MathSciNet review: 3378852
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Abstract: The problem of looking for subsets of the natural numbers which contain no three-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. This construction is optimal in the sense that our method gives a way of effectively computing the greatest possible upper density of a geometric-progression-free set. We also show that geometric progressions in $ \mathbb{Z}/n\mathbb{Z}$ behave more like Roth's theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of $ n$ while avoiding geometric progressions.


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Additional Information

Nathan McNew
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Email: nathan.g.mcnew.gr@dartmouth.edu

DOI: https://doi.org/10.1090/mcom/2979
Received by editor(s): October 8, 2013
Received by editor(s) in revised form: March 1, 2014
Published electronically: May 14, 2015
Article copyright: © Copyright 2015 American Mathematical Society