Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(a_n)_{n \in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish asymptotic upper bounds for the maximal gap in the set of dilates <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{\varvec{\alpha }a_n \}_{n \le N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≤</mo> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> modulo 1 as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, for Lebesgue–almost all dilation vectors <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varvec{\alpha }\in [0,1]^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. More precisely, we prove that for any lacunary <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(a_n)_{n \in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and Lebesgue–almost all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varvec{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> </math></EquationSource> </InlineEquation>, every convex set in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$[0,1]^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> of volume at least <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\log N)^{2}/N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> must contain an element of the set <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\{\varvec{\alpha }a_n \}_{n \le N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <mrow> <mi mathvariant="bold-italic">α</mi> </mrow> <msub> <mi>a</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>≤</mo> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> mod 1, for all sufficiently large N. We also establish a generalized version of this result, where the d-dimensional Lebesgue measure is replaced by a general measure satisfying a certain Fourier decay condition. Our result is optimal up to logarithmic factors, and recovers as a special case a recent result for dimension <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence

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2025年9月18日

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Eduard Stefanescu

摘要

Let $(a_n)_{n \in \mathbb {N}}$ ${(a_{n})}_{n \in N}$ be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \ge 1$ $d \geq 1$ , we establish asymptotic upper bounds for the maximal gap in the set of dilates $\{\varvec{\alpha }a_n \}_{n \le N}$ ${α a_{n}}_{n \leq N}$ modulo 1 as $N \rightarrow \infty $ $N \to \infty$ , for Lebesgue–almost all dilation vectors $\varvec{\alpha }\in [0,1]^d$ $α \in {[0, 1]}^{d}$ . More precisely, we prove that for any lacunary $(a_n)_{n \in \mathbb {N}}$ ${(a_{n})}_{n \in N}$ and Lebesgue–almost all $\varvec{\alpha }$ $α$ , every convex set in $[0,1]^d$ ${[0, 1]}^{d}$ of volume at least $(\log N)^{2}/N$ ${(log N)}^{2} / N$ must contain an element of the set $\{\varvec{\alpha }a_n \}_{n \le N}$ ${α a_{n}}_{n \leq N}$ mod 1, for all sufficiently large N. We also establish a generalized version of this result, where the d-dimensional Lebesgue measure is replaced by a general measure satisfying a certain Fourier decay condition. Our result is optimal up to logarithmic factors, and recovers as a special case a recent result for dimension $d=1$ $d = 1$ .