Perfectly packing a square by squares of sidelength f(n)^{-t}
Perfectly packing a square by squares of sidelength f(n)^{-t}
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用边长为 f(n)^{-t} 的正方形完美地包装一个正方形
DOI：
10.1016/j.disc.2022.113293
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发表时间：
2023
影响因子：
0.8
通讯作者：
Keiju Sono
中科院分区：
文献类型：
--
作者：
Kazuya Aokage;Eriko Shinkawa and Hiro-Fumi Yamada;山崎義徳;Keiju Sono
本文证明了对任意1/2< t< 1，存在一个依赖于t的正整数N 0，使得对任意n 0≥ N 0，边长为f（n）-t（n≥ n 0）的平方可以用不相交的内部填充成面积为∑ n= n 0∞ f（n）-2 t的平方，如果函数f满足适当的条件.主要定理（定理1.1）是[15]中Tao定理的推广，该定理证明了f（n）= n的情形。作为推论，我们证明了当f（n）表示算术级数或素数集合的第n个元素时，存在这样的平方填充。在这些情况下，我们给出了N 0关于t的有效下界。此外，我们考虑了f（n）表示孪生素数集合的第n个元素的情况，并证明了边长为f（n）− t（n≥ n 0）的平方可以用不相交的内部填充成比理论预期稍大的平方。
In this paper, we prove that for any 1/2< t< 1, there exists a positive integer N 0 depending on t such that for any n 0≥ N 0, squares of sidelength f (n)− t for n≥ n 0 can be packed with disjoint interiors into a square of area∑ n= n 0∞ f (n)− 2 t, if the function f satisfies some suitable conditions. The main theorem (Theorem 1.1) is a generalization of Tao's theorem in [15], which argued the case f (n)= n. As corollaries, we prove that there are such packings of squares when f (n) represents the nth element of either an arithmetic progression or the set of prime numbers. In these cases, we give effective lower bounds for N 0 with respect to t. Furthermore, we consider the case that f (n) represents the nth element of the set of twin primes and prove that squares of sidelength f (n)− t for n≥ n 0 can be packed with disjoint interiors into a slightly larger square than theoretically expected.