کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4653305 1632763 2016 14 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Large subsets of discrete hypersurfaces in Zd contain arbitrarily many collinear points
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات گسسته و ترکیبات
پیش نمایش صفحه اول مقاله
Large subsets of discrete hypersurfaces in Zd contain arbitrarily many collinear points
چکیده انگلیسی

In 1977 L.T. Ramsey showed that any sequence in Z2 with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average.We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem. Let  d∈N, let  f:Zd→Zd+1be a Lipschitz map and let  A⊂Zdhave positive upper Banach density. Then  f(A)f(A)contains arbitrarily many collinear points.Note that Pomerance’s theorem corresponds to the special case d=1d=1. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher’s theorem.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: European Journal of Combinatorics - Volume 54, May 2016, Pages 163–176
نویسندگان
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