Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be an irrational number and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> a positive integer. Let f(x) be a polynomial with positive integer coefficients. Solving a 2001 problem of Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence A with density <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{k}-\frac{1}{k\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ11"> <EquationSource Format="TEX">$ \big \{f(a_1)+\ldots +f(a_k): a_i\in A, 1\le i\le k\big \}\cap \big \{\lfloor n\alpha \rfloor : n\in \mathbb {N}\big \}=\emptyset . $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>k</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> <mo>∩</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <mrow> <mo>⌊</mo> <mi>n</mi> <mi>α</mi> <mo>⌋</mo> </mrow> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">∅</mi> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Hegyvári also proved that the density given by him is optimal for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we show that the density <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{1}{k}-\frac{1}{k\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> given by Hegyvári is actually optimal for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Note on a conjecture of Sárközy on special sequences

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2026年8月1日

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Yuchen Ding,
Huixi Li,
Zihan Zhang

摘要

Let $\alpha >1$ $α > 1$ be an irrational number and $k\ge 2$ $k \geq 2$ a positive integer. Let f(x) be a polynomial with positive integer coefficients. Solving a 2001 problem of Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence A with density $\frac{1}{k}-\frac{1}{k\alpha }$ $\frac{1}{k} - \frac{1}{k α}$ such that $ \big \{f(a_1)+\ldots +f(a_k): a_i\in A, 1\le i\le k\big \}\cap \big \{\lfloor n\alpha \rfloor : n\in \mathbb {N}\big \}=\emptyset . $ ${f (a_{1}) + \dots + f (a_{k}) : a_{i} \in A, 1 \leq i \leq k} \cap {⌊ n α ⌋ : n \in N} = \emptyset .$ Hegyvári also proved that the density given by him is optimal for $k=2$ $k = 2$ . In this article, we show that the density $\frac{1}{k}-\frac{1}{k\alpha }$ $\frac{1}{k} - \frac{1}{k α}$ given by Hegyvári is actually optimal for all $k\ge 2$ $k \geq 2$ .