The metric theory of small gaps for a sequence of real numbers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911282499682304 |
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| author | Mahajan, Jewel |
| author_facet | Mahajan, Jewel |
| contents | Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{αa_n \text{ mod }1, 1\leq n \leq N\}$ as $N \to \infty$ was initiated by Rudnick, who established that the minimal gap admits an asymptotic upper bound expressible in terms of the additive energy of $\{a_1,\ldots,a_N\}$ for almost every $α$. Later, Aistleitner, El-Baz, and Munsch demonstrated that the metric theory of minimal gaps for such sequences is governed not by the additive energy, but by the cardinality of the difference set of $\{a_1,\ldots,a_N\}$. They established a sharp convergence test for the typical asymptotic order of the minimal gap and proved general upper and lower bounds that are readily applicable. A key element of their proof relies on the resolution of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. In this article, we generalise several results from the article of Aistleitner, El-Baz, and Munsch on \emph{integer} sequences to the case of \emph{real} sequences. While an upper bound for $δ_{\min}^α(N)$ remains elusive, we obtain one for its floored counterpart $\lfloor δ^α_{\min} \rfloor (N)$ for real sequences $(a_n)_{n \geq 1}$ of distinct numbers. Our theorems recover Theorems 1-3, as well as the result from Section 4.3 of the article by Aistleitner, El-Baz, and Munsch. Furthermore, we establish lower bounds for the minimal gaps of well-spaced sequences and, more generally, of a broader family that contains them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The metric theory of small gaps for a sequence of real numbers Mahajan, Jewel Number Theory Primary 11B05, 11J71, 11J83, 11M06, Secondary 11K06, 11K38 Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{αa_n \text{ mod }1, 1\leq n \leq N\}$ as $N \to \infty$ was initiated by Rudnick, who established that the minimal gap admits an asymptotic upper bound expressible in terms of the additive energy of $\{a_1,\ldots,a_N\}$ for almost every $α$. Later, Aistleitner, El-Baz, and Munsch demonstrated that the metric theory of minimal gaps for such sequences is governed not by the additive energy, but by the cardinality of the difference set of $\{a_1,\ldots,a_N\}$. They established a sharp convergence test for the typical asymptotic order of the minimal gap and proved general upper and lower bounds that are readily applicable. A key element of their proof relies on the resolution of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. In this article, we generalise several results from the article of Aistleitner, El-Baz, and Munsch on \emph{integer} sequences to the case of \emph{real} sequences. While an upper bound for $δ_{\min}^α(N)$ remains elusive, we obtain one for its floored counterpart $\lfloor δ^α_{\min} \rfloor (N)$ for real sequences $(a_n)_{n \geq 1}$ of distinct numbers. Our theorems recover Theorems 1-3, as well as the result from Section 4.3 of the article by Aistleitner, El-Baz, and Munsch. Furthermore, we establish lower bounds for the minimal gaps of well-spaced sequences and, more generally, of a broader family that contains them. |
| title | The metric theory of small gaps for a sequence of real numbers |
| topic | Number Theory Primary 11B05, 11J71, 11J83, 11M06, Secondary 11K06, 11K38 |
| url | https://arxiv.org/abs/2511.16647 |