Quantitative lower bound for Pythagorean pairs in boxes of prime powers

Source abstract

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in\{1,\ldots,N\}$, and set $$Φ_N=\left\{\prod_{i=1}^{N}p_i^{a_i}:a_i\in\{0,1,\ldots,N-1\}\right\}.$$ Write $f(N)$ for the largest cardinality of a subset of $Φ_N$ containing no two distinct legs of an integral right triangle. We prove $$N^{N-1}(\log\log\log N)\ll f(N).$$