On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914001416355840 |
|---|---|
| author | Stefanescu, Eduard |
| author_facet | Stefanescu, Eduard |
| contents | Let \((a_n)_{n \in \mathbb{N}}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates \(\{\boldsymbolα a_n \}_{n \leq N}\) modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbolα \in [0,1]^d$. More precisely, we prove that for any lacunary \((a_n)_{n \in \mathbb{N}}\) and Lebesgue--almost all $\boldsymbolα$, every convex set in $[0,1]^d$ of volume at least $(\log N)^{2+\varepsilon}/N$ must contain an element of the set \(\{\boldsymbolα a_n \}_{n \leq N}\) 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 $d=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence Stefanescu, Eduard Number Theory 11J83, 11J71, 11J70, 42A16, 28A78 Let \((a_n)_{n \in \mathbb{N}}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates \(\{\boldsymbolα a_n \}_{n \leq N}\) modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbolα \in [0,1]^d$. More precisely, we prove that for any lacunary \((a_n)_{n \in \mathbb{N}}\) and Lebesgue--almost all $\boldsymbolα$, every convex set in $[0,1]^d$ of volume at least $(\log N)^{2+\varepsilon}/N$ must contain an element of the set \(\{\boldsymbolα a_n \}_{n \leq N}\) 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 $d=1$. |
| title | On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence |
| topic | Number Theory 11J83, 11J71, 11J70, 42A16, 28A78 |
| url | https://arxiv.org/abs/2504.03575 |