The distribution of $\alpha p$ modulo one in real quadratic function fields

Esrafil Ali Molla

REVIEW

The distribution of $\alpha p$ modulo one in real quadratic function fields

Esrafil Ali Molla

Number Theory · Submitted 16 Oct 2026

Pith's verdict: Solid, well-executed paper: extends Matomäki's theorem to all real quadratic function fields unconditionally, using a finite infinity-type trick that sidesteps truncation errors.

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arXiv ID2610.23742 (version 2)

Pith IDY6LK54FR, Pith's permanent record of this paper and its review

CitationEsrafil Ali Molla. "The distribution of $\alpha p$ modulo one in real quadratic function fields." arXiv:2610.23742 (2026). https://pith.science/paper/Y6LK54FR

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Editor's note

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This paper does exactly what it says: proves the function-field analogue of Matomäki's exponent 1/3 for Diophantine approximation with prime denominators, now covering all real quadratic extensions of Fq(T) unconditionally. The result is new — the real quadratic case was open — and together with the authors' earlier work completes the quadratic function field program.

What makes the paper work is a clean observation: in real quadratic function fields, the infinity types of Hecke characters form a finite set (Corollary 3.8), parametrized by n mod 2R. This lets the author write the Dirichlet character value at a prime element as an exact, finite linear combination of Hecke character values (Proposition 6.3), with no truncation error. That's the main technical novelty, and it's handled well. The supporting material — the explicit Hecke character description in Section 3, the counting lemmas in Section 5, the main estimate in Section 6 — is self-contained, clearly explained, and correctly reasoned. The Riemann hypothesis for function fields (Weil) makes everything unconditional, which is the point.

Soft spots are few and minor. The proof leans on the Dirichlet approximation theorem (Theorem 4.1, cited from Ganguly/Thunder), which is a published result applied within its stated hypotheses; that's fine. The Hecke character section is thorough but may feel heavy to readers who only need the end product. The counting lemmas are elementary but do the job. I don't see any logical gaps or hidden assumptions. The result is a natural analogue of the number field case, not a paradigm shift, but the method for handling the infinite unit group is genuinely new.

The paper deserves a serious referee. It is a genuine contribution to analytic number theory over function fields, and the arguments hold up. I would cite it if working in that area and would bring it to a reading group focused on Diophantine approximation in positive characteristic.

Referee report

Significance

The result is the first unconditional analogue of Matomäki's exponent θ = 1/3 for all real quadratic function fields, removing previous restrictions on class number and on q (q odd). The proof introduces an explicit description of Hecke characters of non-trivial infinity type for real quadratic function fields (Section 3), leading to a finite set of such characters (Corollary 3.8) and an exact Fourier decomposition (Proposition 6.3). This avoids the truncation errors that appear in the number-field analogue [2]. The exposition is clear, the counting lemmas are elementary, and the analytic input (Weil's Riemann hypothesis) is a known theorem, so the result is unconditional. The paper is a significant contribution to Diophantine approximation in function fields.

Minor objections (3)

[Abstract] The citation label \cite{BaMo} does not appear in the reference list; the corresponding entry is [5] (Baier, Molla, Ganguly). Please ensure consistency between the citation key and the bibliography.
[Section 3.3, formula (3.14)] The description of the extension of Hecke characters using m_l-th roots and the symbolic exponent j_l/m_l could be misinterpreted if the m_l are not divisors. The meaning is clear from context, but a short clarifying sentence (e.g., 'one fixes a choice of an m_l-th root of χ_Hecke(a_l^{m_l}) and raises it to the j_l-th power') would improve readability.
[Lemma 3.18] In the proof, the symbol f is used for both the ideal and an element of that ideal. The sentence 'let f ∈ O_K be a generator of f iff f is principal, or otherwise any non-zero element of f' is understandable but slightly informal. Replacing 'f' for the element by a different letter (e.g., g) would avoid overloaded notation.

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The paper

The paper claims

Unconditional analogue of Matomäki's exponent 1/3 for all real quadratic function fields.

The reading

Matomäki proved that for any irrational real α, there are infinitely many primes p with |α - a/p| ≤ $p^{{-4/3+ε}}$. This paper establishes the function-field analogue of Matomäki's exponent θ = 1/3 for every real quadratic extension of F_q(T), unconditional because the Riemann hypothesis for Hecke L-functions is known in the function-field setting. The main theorem gives, for any pair (x_1, x_2) not coming from the field itself, infinitely many principal prime ideals whose ratios approximate the pair with error at most N(p)^{-2/3+ε}. Together with the author's earlier work on rational and imaginary quadratic function fields, this completes the quadratic case.

What carries the argument

The central machinery is the explicit description of Hecke characters of non-trivial infinity type for real quadratic function fields. For a real quadratic extension, the group at infinity is two copies of k_∞ ≅ F_q((1/T)). A continuous character of k_∞^× factors through sign, valuation, and a principal-unit part. The compatibility with the infinite unit group forces the parameter v to be of the form π(n+γ)/(R log q), indexed by n modulo 2R where R is the regulator. Thus, up to norm twists, there are only finitely many infinity types for a given Dirichlet character (Corollary 3.8). This finiteness allows the discrete Fourier inversion (Proposition 6.3) that expresses χ(p) as an exact sum over 2R Hecke characters, with coefficients given by a 2R-point discrete Fourier transform. The Hecke L-functions satisfy the Riemann hypothesis by Weil, and counting lemmas in the ultrametric setting handle the main terms and error terms.

Extended reading notes

Core claim

The paper proves that for any real quadratic function field K over F_q(T) (q odd) and any pair (x_1, x_2) in k_∞^2 not lying in the diagonal embedding of K, there exist infinitely many principal prime ideals p of O_K and elements b ∈ O_K such that |x_i - σ_i(b)/σ_i(π)| ≤ N(p)^{-2/3+ε} for i=1,2, where π is a suitably normalised generator of p. This is the exact analogue of Matomäki's classical exponent θ=1/3. The result is unconditional because the Riemann hypothesis for Hecke L-functions holds for function fields (Weil). The proof overcomes the obstacle of infinitely many units in the real quadratic case by using Hecke characters of non-trivial infinity type, whose infinity types form a finite set (parametrised by n modulo the regulator R). Consequently, the Fourier expansion of a Dirichlet character on normalised prime generators becomes an exact finite linear combination of Hecke character values, avoiding the truncation errors that occur in the number-field analogue.

Load-bearing premise

The proof relies on the existence of infinitely many approximations (a_n, c_n) satisfying the Dirichlet approximation bound (Theorem 4.1) and such that N(f_n) goes to infinity; if this Dirichlet approximation theorem were false or inapplicable for the given pair (x_1, x_2), the counting argument would have no starting point.

Editorial extensions

If this is right

The result, combined with earlier work on rational and imaginary quadratic function fields, completes the analogue of Matomäki's theorem for all quadratic extensions of F_q(T).
It provides an unconditional proof of the exponent θ=1/3, whereas the number-field analogue requires the generalised Riemann hypothesis.
The explicit description of Hecke characters and the exact finite Fourier decomposition may be applicable to other Diophantine approximation problems in function fields where infinite units pose an obstacle.
The method removes the restrictions on class number and on q (apart from q odd) present in earlier work that only achieved the weaker Vaughan exponent 1/4.
The quantitative bound N(p)^{-2/3+ε} is the best that current technology can achieve; any improvement would require new ideas beyond the Riemann hypothesis for Hecke L-functions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The finiteness of infinity types (only R possibilities modulo norm twists) reflects the fact that the regulator R controls the spacing of the degrees of σ_1(π)/σ_2(π) for normalised prime generators, giving a clean conceptual simplification over the number-field case where truncation is needed.
The unconditional nature highlights a structural advantage of function fields: the Riemann hypothesis for Hecke L-functions is known globally, suggesting that function fields can serve as a testing ground for unconditional approaches pursued conditionally in number fields.
The method may extend to higher-degree extensions of function fields, provided one can classify the infinity types of the relevant Hecke characters, which is a purely local algebraic problem.

Original abstract

In the thirties of the last century, I.\,M. Vinogradov proved that the inequality $\|p\alpha\|\le p^{-1/5+\varepsilon}$ has infinitely many prime solutions $p$, where $\|\cdot\|$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$, using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matom\"aki (2009), who showed that the inequality $\|p\alpha\|\le p^{-1/3+\varepsilon}$ has infinitely many prime solutions. This exponent $1/3$ is considered the limit of current technology. Baier and the author established an analogue of Matom\"aki's result for the function field $k=\Fq(T)$ and its imaginary quadratic extensions. Moreover, for real quadratic extensions of $k$ of class number one, they obtained an analogue of Vaughan's result (exponent $1/4$). In this paper, we establish the analogue of Matom\"aki's result for all real quadratic extensions of $k$ without any restriction on the class number.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 2,963 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Matom\"aki proved that if $\alpha\in\R$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer $a$. In \cite{BaMo}, Baier and the author established analogues of this result for $\F_q(T)$ and its imaginary quadratic extensions. In this paper, we extend it to all real quadratic extensions of $\F_q(T)$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Matom\"aki proved that if $\alpha\in\R$ is irrational, then there are infinitely many primes $p$ such that $\|\alpha-a/p\|\le p^{-4/3+\varepsilon}$ for a suitable integer $a$.	background 1.00	0.02	0.03	asserted 1.00
s1	In \cite{BaMo}, Baier and the author established analogues of this result for $\F_q(T)$ and its imaginary quadratic extensions.	background 0.89	0.79	0.04	asserted 1.00
s2	In this paper, we extend it to all real quadratic extensions of $\F_q(T)$.	contribution 1.00	0.81	0.74	asserted 0.94

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.91.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Discussion

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