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<p>For any set <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> <EquationSource Format="TEX">$A$</EquationSource> </InlineEquation> of natural numbers with positive upper Banach density and any <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>k</mi> <mo>⩾</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$k\geqslant 1$</EquationSource> </InlineEquation>, we show the existence of an infinite set <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>B</mi> <mo>⊂</mo> <mi mathvariant="double-struck">N</mi> </math></EquationSource> <EquationSource Format="TEX">$B\subset \mathbb{N}$</EquationSource> </InlineEquation> and a shift <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>t</mi> <mo>⩾</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$t\geqslant 0$</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>A</mi> <mo>−</mo> <mi>t</mi> </math></EquationSource> <EquationSource Format="TEX">$A-t$</EquationSource> </InlineEquation> contains all sums of <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>m</mi> </math></EquationSource> <EquationSource Format="TEX">$m$</EquationSource> </InlineEquation> distinct elements from <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>B</mi> </math></EquationSource> <EquationSource Format="TEX">$B$</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>m</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$m\in \{1,\ldots ,k\}$</EquationSource> </InlineEquation>. This can be viewed as a density analog of Hindman’s finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.</p>

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The density finite sums theorem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月24日

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  • Bryna Kra,
  • Joel Moreira,
  • Florian K. Richter,
  • Donald Robertson

摘要

For any set A $A$ of natural numbers with positive upper Banach density and any k ⩾ 1 $k\geqslant 1$ , we show the existence of an infinite set B ⊂ N $B\subset \mathbb{N}$ and a shift t ⩾ 0 $t\geqslant 0$ such that A − t $A-t$ contains all sums of m $m$ distinct elements from B $B$ for all m ∈ { 1 , … , k } $m\in \{1,\ldots ,k\}$ . This can be viewed as a density analog of Hindman’s finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.