Perfectly packing a square by squares of sidelength $f(n{)}^{-t}$

Abstract

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 $n_0\ge N_0$, squares of sidelength $f(n{)}^{-t}$ for $n\ge n_0$ 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 $n\ge n_0$ can be packed with disjoint interiors into a slightly larger square than theoretically expected.

23 pages. 1 figure