Evenly divisible rational approximations of quadratic irrationalities

Carmon, Dan

doi:10.1007/s11856-017-1624-6

Cited by 2 publications

(1 citation statement)

References 2 publications

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“…They proved asymptotic upper and lower bounds for δ (α) min (N) assuming that α satisfies certain Diophantine approximation properties; for example they established results for certain quadratic irrationals, which have since been extended to all positive quadratic irrationals by Carmon [10], and for algebraic irrationals of higher degree. We refer to [7] for the precise statement of their results.…”

Section: Remarkmentioning

confidence: 99%

Difference sets and the metric theory of small gaps

Aistleitner¹,

El-Baz²,

Munsch³

2021

Preprint

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Let (a n ) n≥1 be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of {a n α mod 1, 1 ≤ n ≤ N } as N → ∞, valid for Lebesgue-almost all α and formulated in terms of the additive energy of {a 1 , . . . , a N }. In the present paper we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of {a 1 , . . . , a N }. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap, and prove (slightly weaker) general upper and lower bounds which allow for a direct application. A major input for these results comes from the recent proof of the Duffin-Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick and Radziwi l l on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

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Section: Remarkmentioning

confidence: 99%

Difference sets and the metric theory of small gaps

Aistleitner¹,

El-Baz²,

Munsch³

2021

Preprint

View full text Add to dashboard Cite

show abstract

Difference Sets and the Metric Theory of Small Gaps

Aistleitner

El-Baz

Munsch

2021

International Mathematics Research Notices

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Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper, Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \mod 1, ~1 \leq n \leq N\}$ as $N \to \infty $, valid for Lebesgue-almost all $\alpha $ and formulated in terms of the additive energy of $\{a_1, \dots , a_N\}$. In the present paper, we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots , a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap and prove (slightly weaker) general upper and lower bounds that allow for a direct application. A major input for these results comes from the recent proof of the Duffin–Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick, and Radziwiłł on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

show abstract

Evenly divisible rational approximations of quadratic irrationalities

Cited by 2 publications

References 2 publications

Difference sets and the metric theory of small gaps

Difference sets and the metric theory of small gaps

Difference Sets and the Metric Theory of Small Gaps

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