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 N0 depending on t such that for any n0≥N0, squares of sidelength f(n)−t for n≥n0 can be packed with disjoint interiors into a square of area ∑n=n0∞f(n)−2t, 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 N0 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≥n0 can be packed with disjoint interiors into a slightly larger square than theoretically expected.