科研之友
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摘要
Let T(n) denote the maximum number of unit distances that a set of n points in the Euclidean plane R-2 can determine with the additional condition that the distinct unit length directions determined by the configuration must be Q-independent. This is related to the Erdos unit distance problem but with a simplifying additional assumption on the direction set that holds "generically". We show that T(n + 1) - T(n) is the Hamming weight of n, i.e. the number of nonzero binary coefficients in the binary expansion of n, and find a formula for T(n) explicitly. In particular, T(n) is Theta(nlog(n)). Furthermore, we describe a process to construct a set of n points in the plane with Q-independent unit length direction set that achieves exactly T(n) unit distances. In the process of doing this, we show T(n) is also the same as the maximum number of edges a subset of vertices of size n determines in either the countably infinite lattice Z(infinity) or the infinite hypercube graph {0,1}(infinity). The problem of determining T(n) can be viewed as either a type of packing or isoperimetric problem.
- 出版日期2015
