کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4653305 | 1632763 | 2016 | 14 صفحه PDF | دانلود رایگان |

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.
Journal: European Journal of Combinatorics - Volume 54, May 2016, Pages 163–176