Approximate polynomial structure in additively large sets

Creators: Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl

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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