<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> denote the Liouville function. We show that the logarithmic mean of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="131" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mi>n</mi> <mo>⌋</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mi>n</mi> <mo>⌋</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is 0 whenever <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _1,\alpha _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are positive reals with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _1/\alpha _2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo stretchy="false">/</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> irrational. We also show that for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\geqslant 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> the logarithmic mean of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mi>n</mi> <mo>⌋</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>⋯</mo> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mo>⌊</mo> <msub> <mi>α</mi> <mi>k</mi> </msub> <mi>n</mi> <mo>⌋</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3770_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha _i.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.</p>

On a Bohr set analogue of Chowla’s conjecture

期刊论文
出版模式：

开放获取
发表日期：

2025年5月28日

查看全文

Joni Teräväinen,
Aled Walker

摘要

Let $\lambda $ $λ$ denote the Liouville function. We show that the logarithmic mean of $\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$ $λ (⌊ α_{1} n ⌋) λ (⌊ α_{2} n ⌋)$ is 0 whenever $\alpha _1,\alpha _2$ $α_{1}, α_{2}$ are positive reals with $\alpha _1/\alpha _2$ $α_{1} / α_{2}$ irrational. We also show that for $k\geqslant 3$ $k ⩾ 3$ the logarithmic mean of $\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$ $λ (⌊ α_{1} n ⌋) \dots λ (⌊ α_{k} n ⌋)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $\alpha _i.$ $α_{i} .$ Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crnčević–Hernández–Rizk–Sereesuchart–Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.