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<p>We prove that for any finite partition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {Z}=\cup ^{r}_{i=1} A_i\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo>=</mo> <msubsup> <mo>∪</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>r</mi> </msubsup> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, there exists <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(i\in \{1,\dots ,r\} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>r</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> such that for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(s_1, s_2 \in \mathbb {Z}^+\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, the set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(s_1A_i-s_1A_i+s_2A_i\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> contains a Bohr set. This is a generalization of a result of Anh Le and the first author.</p>

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On Bohr Sets Arising from Partitions of \(\mathbb {Z}\)

  • 期刊论文

  • 发表日期：

    2025年7月21日

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  • Thái Hoàng Lê,
  • Gauree Wathodkar

摘要

We prove that for any finite partition \(\mathbb {Z}=\cup ^{r}_{i=1} A_i\) Z = ∪ i = 1 r A i , there exists \(i\in \{1,\dots ,r\} \) i ∈ { 1 , ⋯ , r } such that for all \(s_1, s_2 \in \mathbb {Z}^+\) s 1 , s 2 ∈ Z + , the set \(s_1A_i-s_1A_i+s_2A_i\) s 1 A i - s 1 A i + s 2 A i contains a Bohr set. This is a generalization of a result of Anh Le and the first author.