论文标题
完美地包装一个正方形的正方形$ f（n）^{ - t} $
Perfectly packing a square by squares of sidelength $f(n)^{-t}$
论文作者
论文摘要
在本文中，我们证明，对于任何$ 1/2 <t <1 $，都存在一个正整数$ n_ {0} $，具体取决于$ t $，以至于对于任何$ n_ {0} \ geq n_ {0} $，sideLegenth $ f（n）^n）^{ - t} $ for $ n _ geq n _ square for $ N_如果函数$ f $满足某些合适的条件，则面积$ \ sum_ {n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = n = sum_ {\ infty} f（n）^{ - 2t} $。主要定理（定理1.1）是对道的定理的概括，该概述了$ f（n）= n $的情况。作为推论，我们证明，当$ f（n）$代表算术进展或质数集的$ n $ th元素时，有一些正方形包装。在这些情况下，我们相对于$ t $提供了$ n_ {0} $的有效下限。此外，我们认为$ f（n）$代表了双素数的$ n $ th元素，并证明sideLength $ f（n）^{ - t} $的平方对于$ n \ geq n_ {0} $可以包装成比稍大的正方形，比稍大于理论上的广场。
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 $n_{0}\geq N_{0}$, squares of sidelength $f(n)^{-t}$ for $n\geq n_{0}$ can be packed with disjoint interiors into a square of area $\sum_{n=n_{0}}^{\infty}f(n)^{-2t}$, if the function $f$ satisfies some suitable conditions. The main theorem (Theorem 1.1) is a generalization of Tao's theorem, which argued the case $f(n)=n$. As corollaries, we prove that there are such packings of squares when $f(n)$ represents the $n$th 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 $n$th element of the set of twin primes and prove that squares of sidelength $f(n)^{-t}$ for $n\geq n_{0}$ can be packed with disjoint interiors into a slightly larger square than theoretically expected.
