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Published 2016 | Version Submitted + Published
Journal Article Open

Approximate polynomial structure in additively large sets

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Abstract

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explored.

Additional Information

© 2016 The Aurhors. Received: 8/10/15, Accepted: 6/15/16, Published: 7/7/16. I. Goldbring was partially supported by NSF CAREER grant DMS-1349399. M. Lupini was supported by the York University Susan Mann Dissertation Scholarship and by the ERC Starting grant no. 259527 of Goulnara Arzhantseva. K. Mahlburg was supported by NSF Grant DMS-1201435. This work was initiated during a week-long meeting at the American Institute for Mathematics on August 4-8, 2014 as part of the SQuaRE (Structured Quartet Research Ensemble) project "Nonstandard Methods in Number Theory." The authors would like to thank the Institute for the opportunity and for the Institute's hospitality during their stay.

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Published - q49.pdf

Submitted - 1508.02350v1.pdf

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1508.02350v1.pdf

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Additional details

Identifiers

Eprint ID
70793
Resolver ID
CaltechAUTHORS:20161004-084936564

Related works

Describes
https://arxiv.org/abs/1508.02350 (URL)

Funding

NSF
DMS-1349399
York University
European Research Council (ERC)
259527
NSF
DMS-1201435

Dates

Created
2016-10-04
Created from EPrint's datestamp field
Updated
2023-06-02
Created from EPrint's last_modified field