Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers
Chi Hoi Yip
TL;DR
For $k\ge3$ and nonzero $n$, the paper proves that perturbing $o(X^{1/k})$ elements of the set $\{x^k+n:x\in\mathbb N\}$ up to $X$ preserves multiplicative irreducibility, extending previous results for $M_k'$. The proof hinges on a local-bipartite-Diophantine-tuple framework and extremal graph theory (Kővári–Sós–Turán) to rule out multiplicative decompositions by showing any such decomposition would force forbidden local structures. Key ingredients include a gap principle for simultaneous $k$-th power representations and sharp bounds on the number of large elements in one side of a BD$_k(n)$ tuple. The result advances understanding of how small perturbations interact with multiplicative structure and highlights connections between Diophantine tuples, extremal graph theory, and additive-multiplicative decompositions.
Abstract
Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted $k$-th powers. They conjectured that for each $k\geq 2$, if one changes $o(X^{1/k})$ elements of $M_k'=\{x^k+1: x \in \mathbb{N}\}$ up to $X$, then the resulting set cannot be written as a product set $AB$ nontrivially. In this paper, we confirm a more general version of their conjecture for $k\geq 3$.