Pythagorean pairs in boxes of prime powers

Abstract

Let $p_i$ be the $i$th prime, and let $f(N)$ be the largest cardinality of a subset of \[ \left\{\prod_{i=1}^N p_i^{a_i}:0\le a_i<N,\ a_i\in\mathbb{Z}\right\} \] containing no two distinct legs of an integer right triangle. We prove \[ f(N) \ll N^N (\log\log N)^{-1/1296}. \]