کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4593635 1630663 2015 14 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Geometric progression-free sequences with small gaps
ترجمه فارسی عنوان
توالی هندسی بدون پیشرفت با شکاف های کوچک
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression-free sequence of positive integers. In this paper we prove the existence of geometric progression-free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence T   not containing any 6-term geometric progressions such that for any x≥1x≥1 and ε>0ε>0 the interval [x,x+Cεexp⁡((C+ε)log⁡x/log⁡log⁡x)][x,x+Cεexp⁡((C+ε)log⁡x/log⁡log⁡x)] contains an element of T  , where C=56log⁡2 and Cε>0Cε>0 is a constant depending on ε  . As an intermediate result we prove a bound on sums of functions of the form f(n)=exp⁡(−dk(n))f(n)=exp⁡(−dk(n)) in very short intervals, where dk(n)dk(n) is the number of positive k-th powers dividing n, using methods similar to those that Filaseta and Trifonov used to prove bounds on the gaps between k-th power free integers.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 151, June 2015, Pages 197–210
نویسندگان
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