Published December 2015
| Submitted
Journal Article
Open
Sets characterized by missing sums and differences in dilating polytopes
Abstract
A sum-dominant set is a finite set A of integers such that |A+A|>|A−A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of {0,…,n} is bounded below by a positive constant as n→∞. Hegarty then extended their work and showed that for any prescribed s,d ∈ N_0, the proportion P^(a,d)_n of subsets of {0,…,n} that are missing exactly s sums in {0,…,2n} and exactly 2d differences in {−n,…,n} also remains positive in the limit.
Additional Information
© 2015 Elsevier Inc. Received 21 June 2014, Revised 25 January 2015, Accepted 13 April 2015, Available online 29 June 2015. This research was conducted as part of the 2013 SMALL REU program at Williams College and was partially supported funded by NSF grant DMS0850577 and Williams College; the third named author was also partially supported by NSF grant DMS1265673. We would like to thank our colleagues from the Williams College 2013 SMALL REU program, especially Frank Morgan, as well as Kevin O'Bryant for helpful conversations, and Jonas Meyer for supplying the proof of the volume inequality (6.2). We would also like to thank our referee, whose detailed comments greatly improved our exposition at several points throughout this paper, as well as gave us additional interesting questions to consider for future work.Attached Files
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Additional details
- Eprint ID
- 60015
- DOI
- 10.1016/j.jnt.2015.04.027
- Resolver ID
- CaltechAUTHORS:20150901-145143034
- DMS-0850577
- NSF
- Williams College
- DMS-1265673
- NSF
- Created
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2015-09-01Created from EPrint's datestamp field
- Updated
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2021-11-10Created from EPrint's last_modified field