Pythagorean pairs in boxes of prime powers

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