<p>We show that for all <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$A, B \subseteq \{0,1,2\}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊆</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> we have <Equation ID="Equ37"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_Equ37.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </MediaObject> <EquationSource Format="TEX">$|A+B|\geqslant (|A||B|)^\frac{\log 5}{2\log 3}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">|</mo> </mrow> <mo>⩾</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mfrac> <mrow> <mo>log</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>log</mo> <mn>3</mn> </mrow> </mfrac> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We also show that for all finite <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$A,B \subset \mathbb {Z}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and any <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$V \subseteq \{0,1\}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>⊆</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> the inequality <Equation ID="Equ38"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_Equ38.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="292" /> </MediaObject> <EquationSource Format="TEX">$|A+B+V|\geqslant |A|^{1/p}|B|^{1/q}|V|^{\log _{2}(p^{1/p}q^{1/q})}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">|</mo> </mrow> <mo>⩾</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>B</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>q</mi> </mrow> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>V</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <msub> <mo>log</mo> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> </msup> <msup> <mi>q</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mrow> </math></EquationSource> </Equation>holds for all <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$p \in (1, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_180_Article_IEq5.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$q=\frac{p}{p-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mfrac> <mi>p</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is the conjugate exponent of <i>p</i>. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems.</p>

Discrete Brunn–Minkowski inequality for subsets of the cube

期刊论文
发表日期：

2025年9月25日

查看全文

Lars Becker,
Paata Ivanisvili,
Dmitry Krachun,
José Madrid

摘要

We show that for all $A, B \subseteq \{0,1,2\}^{d}$ $A, B \subseteq {0, 1, 2}^{d}$ we have $|A+B|\geqslant (|A||B|)^\frac{\log 5}{2\log 3}.$ $| A + B | ⩾ {(| A | | B |)}^{\frac{log 5}{2 log 3}} .$ We also show that for all finite $A,B \subset \mathbb {Z}^{d}$ $A, B \subset Z^{d}$ , and any $V \subseteq \{0,1\}^{d}$ $V \subseteq {0, 1}^{d}$ the inequality $|A+B+V|\geqslant |A|^{1/p}|B|^{1/q}|V|^{\log _{2}(p^{1/p}q^{1/q})}$ $| A + B + V | ⩾ {| A |}^{1 / p} {| B |}^{1 / q} {| V |}^{{log}_{2} (p^{1 / p} q^{1 / q})}$ holds for all $p \in (1, \infty )$ $p \in (1, \infty)$ , where $q=\frac{p}{p-1}$ $q = \frac{p}{p - 1}$ is the conjugate exponent of p. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems.