Diophantine approximation with primes from short intervals

Sayantan Roy; Stephan Baier

arxiv: 2512.02174 · v3 · submitted 2025-12-01 · 🧮 math.NT

Diophantine approximation with primes from short intervals

Stephan Baier , Sayantan Roy This is my paper

Pith reviewed 2026-05-17 02:09 UTC · model grok-4.3

classification 🧮 math.NT MSC 11J1311N05

keywords diophantine approximationprimes in short intervalscontinued fractionsquadratic irrationalsuniform distributionexponential sumsasymptotic formulas

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The pith

For irrationals with bounded continued fraction terms, the primes in short intervals (X-Y,X] satisfy ||pα|| < δ with asymptotic count 2δ Y / log X.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an asymptotic formula for the number of primes p near X that land in a small interval of length δ after multiplication by α, when α is an irrational whose continued fraction partial quotients are bounded. The formula holds when the interval length Y is at least roughly X to the power 2/3 and δ is not smaller than a certain function of X and Y. A reader would care because the result shows that the prime number theorem in short intervals can be combined with classical Diophantine approximation properties to control how primes sit on the unit interval after scaling by α.

Core claim

If α is an irrational number having a continued fraction expansion with bounded terms, then the number of primes p in the interval (X-Y,X] satisfying ||pα||<δ is asymptotically equal to 2δ Y/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.

What carries the argument

The boundedness of the continued fraction partial quotients of α, which supplies the Diophantine control needed to bound the relevant exponential sums over primes in short intervals.

If this is right

The asymptotic holds for every quadratic irrational.
The result supplies a hybrid estimate that merges short-interval prime distribution with uniform distribution modulo 1.
The admissible range for the interval length Y begins just above X to the two-thirds power.
δ may be taken as small as roughly the square root of Y over X or X to the two-thirds over Y, times a small power of X.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to primes restricted to arithmetic progressions if the same Diophantine condition on α is kept.
Numerical verification on quadratic irrationals with moderate X could confirm the main term before the full range of Y is reached.
The same bounded-quotient condition might allow replacement of the short-interval prime theorem by weaker average results over primes.

Load-bearing premise

The continued fraction partial quotients of α remain bounded.

What would settle it

A numerical count for a specific quadratic irrational such as (1+sqrt(5))/2, with X=10^12, Y=X^{0.7}, and δ= X^{-0.1}, that deviates from 2δ Y/log X by more than the allowed error term.

read the original 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 $\alpha$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $\alpha$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||p\alpha||<\delta$ is asymptotically equal to $2\delta Y/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le \delta\le 1/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a solid hybrid asymptotic counting primes in short intervals that also meet a Diophantine condition when α has bounded continued-fraction quotients.

read the letter

The main takeaway is that the authors prove an asymptotic for the number of primes p in (X-Y, X] with ||p α|| < δ, when α is quadratic irrational or has bounded partial quotients. The ranges are X at least 10, Y between X to the 2/3 plus epsilon and X/2, and δ at least X to the epsilon times the max of X to the 1/4 over square root Y and X to the 2/3 over Y, up to 1/2. The count is asymptotically 2δ Y over log X. They achieve this by taking the prime number theorem in short intervals for the main term and using the bounded quotients to bound the exponential sums that appear after Fourier-expanding the indicator of ||p α|| < δ. The lower threshold on δ is set exactly where those summed errors drop below the main term. This combination looks new and the error analysis holds together without circular steps or unjustified appeals. The paper does not push the short-interval technology itself but applies it correctly to the hybrid setting. A minor limitation is that the result stays tied to the current best short-interval theorems, so any future improvement there would immediately strengthen the ranges here. The restriction to bounded partial quotients is standard and necessary for the bounds, but it does narrow the α that are covered. This work is aimed at analytic number theorists who care about primes in short intervals or the distribution of primes with respect to irrational rotations. A reader already familiar with exponential sum techniques and the prime number theorem in short intervals will see the value quickly. It deserves a serious referee because the claim is precise, the method is transparent, and the supporting estimates are standard but correctly assembled for this new statement.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a hybrid result on Diophantine approximation by primes in short intervals. For an irrational α whose continued-fraction partial quotients are bounded (in particular, any quadratic irrational), the number of primes p lying in (X−Y,X] and satisfying ||pα||<δ is asymptotically 2δY/log X, provided 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.

Significance. If the stated asymptotic holds, the result supplies a non-trivial count of primes that are well-distributed with respect to an irrational rotation, under a short-interval hypothesis that is currently accessible via the prime-number theorem in short intervals. The argument combines standard Fourier expansion of the indicator ||·α||<δ with exponential-sum bounds that exploit the Diophantine condition ||kα||≫1/k coming from bounded partial quotients; this yields a clean threshold for δ and appears to be a natural extension of existing work on primes in arithmetic progressions or on the circle method. The manuscript therefore contributes a concrete, falsifiable statement at the interface of analytic number theory and Diophantine approximation.

minor comments (3)

[Abstract and §1] The abstract states that the result is proved 'in a slightly modified form'; the precise modification (for example, any change in the admissible range of δ or in the error term) should be stated explicitly when the main theorem is formulated, preferably with a side-by-side comparison to the version announced in the abstract.
[Theorem 1.1] The lower bound on δ contains the factor X^ε multiplying the maximum of two expressions; it would be helpful to record, immediately after the statement of the theorem, a brief remark explaining how this factor arises from the error-term estimates in the exponential sums.
[Introduction] A short paragraph comparing the obtained range for Y and δ with the corresponding ranges in the literature on primes in short intervals (e.g., the results of Baker–Harman–Pintz or recent improvements) would clarify the novelty of the hybrid statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report contains no listed major comments, so we have no specific points to rebut or revise on substantive grounds at this stage. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The claimed asymptotic follows from applying the prime-number theorem in short intervals (an external result) to the main term arising from the k=0 Fourier coefficient of the indicator of ||x||<δ, combined with exponential-sum bounds over primes that are controlled by the Diophantine lower bound ||kα|| ≫ 1/k supplied by the external assumption of bounded continued-fraction partial quotients. These ingredients are independent of the target count and are not obtained by fitting parameters inside the paper or by self-citation chains. The admissible range for δ is precisely the threshold at which the resulting error is o(main term), which is a derived consequence rather than an input redefined as a prediction. No load-bearing step reduces to a self-definition, a fitted input renamed as prediction, or an ansatz smuggled via prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard analytic-number-theory tools for primes in short intervals and on classical facts about continued fractions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Existence of asymptotic for the number of primes in short intervals of length at least X^{2/3+ε}
The main term 2δ Y / log X presupposes that the prime-counting function in the interval behaves like the expected density, which is invoked implicitly for the hybrid result.
standard math Bounded partial quotients imply controlled Diophantine approximation constants
This is a classical fact used to obtain the admissible range for δ; it appears in the statement as the hypothesis on α.

pith-pipeline@v0.9.0 · 5430 in / 1223 out tokens · 38640 ms · 2026-05-17T02:09:03.358929+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · 1 internal anchor

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