What question did this study set out to answer?

The aim is to establish results regarding Diophantine approximation with primes in short intervals, particularly involving irrational numbers with bounded continued fractions.

February 22, 2026

Diophantine approximation with primes from short intervals

Key Points

The aim is to establish results regarding Diophantine approximation with primes in short intervals, particularly involving irrational numbers with bounded continued fractions.
Proved results based on properties of irrational numbers and continued fractions.
Analyzed the distribution of primes within specific intervals.
Utilized asymptotic analysis to derive conclusions regarding prime counts.
Found that the count of primes within the specified interval satisfies a specific asymptotic formula.
Identified conditions on the parameters X, Y, and δ for the results to hold.

Abstract

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If α is an irrational number having a continued fraction expansion with bounded terms (in particular, if α is a quadratic irrational), then the number of primes p in the interval (X – Y, X] satisfying ∥pα∥ < δ is asymptotically equal to 2δY/log X, provided that X ≥ 10, X 2/3+ε ≤ Y ≤ X/2 and X ε max X 1/4 Y −1/2, X 2/3 Y −1 ≤ δ ≤ 1/2.

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Cite This Study

Baier et al. (Fri,) studied this question.

synapsesocial.com/papers/699a9d3c482488d673cd3046 https://doi.org/https://doi.org/10.1142/s179304212650065x

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