Multiplicatively reducible subsets of shifted perfect $k$-th powers and bipartite Diophantine tuples

Recently, Hajdu and Sárközy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k\ge 3$, each infinite subset of $\{x^k+1:x\in \mathbb{N}\}$ is multiplicatively irreducible. In this paper, we attempt to make their result effective by building a connection between this problem and the bipartite generalization of the well-studied Diophantine tuples. More precisely, given an integer $k\ge 3$ and a nonzero integer $n$, we call a pair of subsets of positive integers $(A,B)$ a bipartite Diophantine tuple with property $B{D}_k(n)$ if $\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }\ge 2$ and $AB+n\subset \{x^k:x\in \mathbb{N}\}$. We show that $\min \{\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }\}\ll \log \mathrm{\mid }n\mathrm{\mid }$, extending a celebrated work of Bugeaud and Dujella (where they considered the case $n=1$). We also provide an upper bound on $\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }$ in terms of $n$ and $k$ under the assumption $\min \{\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }\}\ge 4$ and $k\ge 6$. Specializing our techniques to Diophantine tuples, we significantly improve several results by Bérczes-Dujella-Hajdu-Luca, Bhattacharjee-Dixit-Saikia, and Dixit-Kim-Murty.