کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4593473 1630657 2015 31 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Sets characterized by missing sums and differences in dilating polytopes
ترجمه فارسی عنوان
مجموعه های مشخص شده توسط مبالغ و تفاوت های گم شده در گسترش چند جمله ای
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

TextA sum-dominant set is a finite set A   of integers such that |A+A|>|A−A||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}{0,…,n} is bounded below by a positive constant as n→∞n→∞. Hegarty then extended their work and showed that for any prescribed s,d∈N0s,d∈N0, the proportion ρns,d of subsets of {0,…,n}{0,…,n} that are missing exactly s   sums in {0,…,2n}{0,…,2n} and exactly 2d   differences in {−n,…,n}{−n,…,n} also remains positive in the limit.We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P   be a polytope in RDRD with vertices in ZDZD, and let ρns,d now denote the proportion of subsets of L(nP)L(nP) that are missing exactly s   sums in L(nP)+L(nP)L(nP)+L(nP) and exactly 2d   differences in L(nP)−L(nP)L(nP)−L(nP). As it turns out, the geometry of P   has a significant effect on the limiting behavior of ρns,d. We define a geometric characteristic of polytopes called local point symmetry, and show that ρns,d is bounded below by a positive constant as n→∞n→∞ if and only if P   is locally point symmetric. We further show that the proportion of subsets in L(nP)L(nP) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P. A direct corollary of these results is that if P   is additionally point symmetric, the proportion of sum-dominant subsets of L(nP)L(nP) also remains positive in the limit.VideoFor a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 157, December 2015, Pages 123–153
نویسندگان
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