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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, if one changes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$o(X^{1/k})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> elements of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M_k'=\{x^k+1: x \in \mathbb {N}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>M</mi> <mi>k</mi> <mo>′</mo> </msubsup> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <msup> <mi>x</mi> <mi>k</mi> </msup> <mo>+</mo> <mn>1</mn> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

Multiplicative irreducibility of small perturbations of the set of shifted k-th powers

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2026年1月7日

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Chi Hoi Yip

摘要

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 $k\ge 2$ $k \geq 2$ , if one changes $o(X^{1/k})$ $o (X^{1 / k})$ elements of $M_k'=\{x^k+1: x \in \mathbb {N}\}$ $M_{k}^{'} = {x^{k} + 1 : x \in N}$ 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 $k\ge 3$ $k \geq 3$ .