<p>In this note, we extend the notion of minimal gaps to higher dimensional sequences. We bound the minimal gap for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\{\varvec{a}_n\varvec{\alpha }\}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mrow> <mi mathvariant="bold-italic">a</mi> </mrow> <mi>n</mi> </msub> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\{a_n\varvec{\alpha }\}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\{\varvec{a}_n\cdot \varvec{\alpha }\})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mrow> <mi mathvariant="bold-italic">a</mi> </mrow> <mi>n</mi> </msub> <mo>·</mo> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in terms of the cardinality of the difference sets of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{a}_n,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">a</mi> </mrow> <mi>n</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a_n^{(1)},\dots ,a_n^{(d)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>a</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msubsup> <mi>a</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation> are sequences of distinct integers.</p>

Minimal gap for higher dimensional sequences

期刊论文
发表日期：

2026年5月30日

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Tanmoy Bera

摘要

In this note, we extend the notion of minimal gaps to higher dimensional sequences. We bound the minimal gap for $(\{\varvec{a}_n\varvec{\alpha }\}),$ $({a_{n} α}),$ $(\{a_n\varvec{\alpha }\}),$ $({a_{n} α}),$ and $(\{\varvec{a}_n\cdot \varvec{\alpha }\})$ $({a_{n} \cdot α})$ in terms of the cardinality of the difference sets of $a_n$ $a_{n}$ and $\varvec{a}_n,$ $a_{n},$ where $a_n$ $a_{n}$ and $a_n^{(1)},\dots ,a_n^{(d)}$ $a_{n}^{(1)}, \dots, a_{n}^{(d)}$ are sequences of distinct integers.