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

TL;DR

This paper generalizes a square packing theorem to functions beyond linear, including primes and twin primes, establishing conditions for packing squares of size f(n)^{-t} into a larger square.

Contribution

It extends Tao's theorem to broader classes of functions, providing new packing results for prime and twin prime sequences with explicit bounds.

Findings

Packings exist for prime and twin prime sequences under certain conditions.

Effective bounds for N_0 depend on the function f and parameter t.

Slightly larger squares can accommodate twin prime-based packings.

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 $n_{0}\geq N_{0}$, 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.