Large dilates of hypercube graphs in the plane

IF 0.6 3区数学 Q3 MATHEMATICS Analysis Mathematica Pub Date : 2024-09-15 DOI:10.1007/s10476-024-00045-6

V. Kovač, B. Predojević

引用次数: 0

Abstract

We study a distance graph $\Gamma_n$ that is isomorphic to the $1$-skeleton of an $n$-dimensional unit hypercube. We show that every measurable set of positive upper Banach density in the plane contains all sufficiently large dilates of $\Gamma_n$. This provides the first examples of distance graphs other than the trees for which a dimensionally sharp embedding in positive density sets is known.

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平面超立方体图的大扩张

我们研究了一个距离图（\γ_n\），它与（n\）维单位超立方体的（1\）骨架同构。我们证明了平面上每一个正上巴纳赫密度的可测集合都包含了 $\Gamma_n$ 的所有足够大的扩张。这提供了除树以外的距离图的第一个例子，已知这些距离图在正密度集合中的维度锐嵌入。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis Mathematica MATHEMATICS-

CiteScore

1.00

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx). The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx). The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.

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