Diophantine approximation with primes in an arithmetic progression

D. Mazumder; J. Sivaraman

REVIEW

Diophantine approximation with primes in an arithmetic progression

D. Mazumder, J. Sivaraman

Number Theory · Submitted 9 Sep 2026 · 9 references

Pith's verdict: Nice extension of Vaughan's result to arithmetic progressions, but the proof of Theorem 1 uses a one-sided Fourier approximation that cannot deliver the two-sided bound needed for Corollary 2.

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arXiv ID2609.10010 (version 1)

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CitationD. Mazumder, J. Sivaraman. "Diophantine approximation with primes in an arithmetic progression." arXiv:2609.10010 (2026). https://pith.science/paper/GZWWCAJ5

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Editor's note

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The paper extends Vaughan's 1977 theorem on Diophantine approximation by primes to the setting of primes in an arithmetic progression, and constructs an uncountable set of irrationals whose convergents have prime denominators satisfying a congruence condition. The second part is novel, self-contained, and addresses an Erdős–Mahler question in a clever way.

The soft spot is in the proof of Theorem 1. Lemma 7 gives only an upper bound for f_Δ(θ) − 2Δ by a sum of exponentials plus an error term. The proof of Theorem 1 uses that one-sided inequality to bound the weighted sum ∑ Λ(n)(f_Δ(αn−β)−2Δ). That is fine as an upper bound. But Corollary 2 converts that weighted sum into an asymptotic formula for the number of primes with ||αp−β|| < Δ, which requires a two-sided estimate — or at least a lower bound of the same quality. An upper bound alone cannot give the claimed asymptotic; it would at best give an upper bound on the prime count. The stress-test note is accurate: the step from Theorem 1 to Corollary 2 implicitly assumes a two-sided bound that was not proved.

This is a load-bearing flaw. The rest of the paper (the construction in Section 4 and the proof that not all numbers in S are Liouville) stands on its own and is interesting. The main analytic part could likely be repaired by using a two-sided Selberg or Fejér kernel approximation for f_Δ, but as written the central result does not follow.

I would send this to a referee. The paper is serious and the authors clearly know the techniques; the gap is real but possibly fixable. The construction in Section 4 is a nice bonus. If the authors can correct the Fourier approximation step, the paper would be a solid contribution.

Referee report

Major objections (2)

[§1, Theorem 1 and Corollary 2] Theorem 1 is proved only as an upper bound: the inequality f_Δ(θ)-2Δ ≤ Σ c_ℓ⁺ e(ℓθ)+1/(H+1) from Lemma 7 gives S = Σ Λ(n)(f_Δ(αn-β)-2Δ) ≤ an error expression. Corollary 2 attempts to use this single bound to obtain an asymptotic for |{p≤N: p≡u mod v, ||αp-β||<Δ}|. The partial‑summation argument leading to the claimed equality (count = 2Δ π(N,u,v) + O(...)) implicitly requires both an upper and a lower bound for S. The proof provides only one direction, so neither the asymptotic nor even the infinitude of such primes (first part of Corollary 2) follows from the given reasoning. The gap is load‑bearing for the central result of the paper.
[§3, Proof of Corollary 2] The deduction of the infinitude claim (first part of Corollary 2) from Theorem 1 with N = q² and Δ = C N^{-1/4} log⁸ N is invalid for the same reason: an upper bound on S cannot force S to be positive. A lower bound (or a two‑sided approximation of f_Δ) is needed. Without it, one cannot conclude that there are any primes with ||αp-β||<Δ beyond those that might be there by coincidence.

Minor objections (5)

[Abstract and §1] In the abstract and again on page 2, the phrase 'primes ℓ such }αℓ−β}' is missing the word 'that' before the braces.
[§1, Theorem 1 statement] The quantity L is defined as log(Nq/Δ) in the abstract but as log(Nq/Δ) in Theorem 1; later in the proof of Theorem 1, L₁ = log(2HN) appears. The notation should be consistently defined at each occurrence.
[§3, proof of Theorem 1, after Eq. (2)] The term N^ϵ q Δ^{1-ϵ} is treated as a separate error term, but its size relative to the other terms is not discussed. Since ϵ is arbitrary, this term is essentially q Δ log⁸(Nq/Δ) after absorbing the ϵ; the authors should clarify whether this is intended to be dominated by the other terms under the chosen parameters.
[§3, proof of Theorem 3] The handling of the congruence condition in the type‑II estimate when N^{1/2} < K < N/U uses Perron's formula (Lemma 6) with a parameter L = N². The choice L = N² seems very large relative to N; the error term should be O(1/N) as claimed, but the size of the integral over |t|≤L might create a factor of log N. A brief justification would help.
[§4, Lemma 9] The definition of irrationality measure in Definition 1 is slightly non‑standard (it uses the exponent in the inequality ||δn|| < 1/n^{λ-1}). The standard definition usually takes the supremum of exponents for which there are infinitely many solutions. This should be aligned or a remark added to explain the difference.

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The paper

The paper claims

For any arithmetic progression, infinitely many primes approximate irrationals to exponent 1/4.

The reading

This paper proves that for any irrational α and any coprime integers u,v, there exist infinitely many primes ℓ ≡ u mod v such that ||αℓ - β|| ≪_v $ℓ^{{-1/4}}$ $log^{8}$ ℓ. This follows from a general theorem (Theorem 1) that extends Vaughan's 1977 estimate for all primes to primes in a fixed arithmetic progression, relying on exponential sum estimates. The paper also constructs an uncountable set S ⊂ ℝ\ℚQ such that every γ ∈ S has infinitely many prime ℓ ≡ u mod v with ||γℓ|| < $ℓ^{{-1}}$, and proves unconditionally that not all elements of S are Liouville numbers, answering a question of Erdős and Mahler from 1939.

What carries the argument

The key machinery is Theorem 1, an estimate for sums Σ_{n≡u mod v} Λ(n)(f_Δ(αn-β)-2Δ) with error terms involving N, q, Δ, whose proof uses exponential sum bounds (Theorem 3) obtained from Vaughan's Type I and Type II estimates after applying Vaughan's identity to the von Mangoldt function. The second part relies on a continued‑fraction construction: the set P of sequences of primes satisfying a recurrence, the definition of S as the set of numbers whose convergents have those denominators, and Lemma 9 (a quantitative irrationality measure criterion) together with Xylouris's theorem on the least prime in an arithmetic progression to bound the growth of denominators and exclude Liouville numbers.

Extended reading notes

Core claim

The central discovery is twofold. First, given coprime u,v, there are infinitely many primes ℓ ≡ u mod v with the Diopantine approximation exponent 1/4 (up to a logarithm), generalising the unconditional result of Vaughan from all primes to a fixed congruence class. Second, an explicit uncountable family of irrationals is produced whose convergents all have denominators that are prime and congruent to u mod v, yielding infinitely many primes ℓ with ||γℓ|| < $ℓ^{{-1}}$, and a criterion based on Xylouris's bound on the least prime in an arithmetic progression shows that this family contains non‑Liouville numbers.

Load-bearing premise

The proof requires that for the given irrational α and integers v,N, there exists a rational approximation a/q with q > $N^{{1/4}}$ and |αv - a/q| ≤ 1/q², which is guaranteed by Dirichlet's theorem but may not hold for every N unless one can always choose a sufficiently refined convergent.

Editorial extensions

If this is right

The Diopantine approximation exponent 1/4 for primes in arithmetic progressions is the best unconditional result available; it improves on previous bounds that applied only to all primes.
The construction yields an uncountable family of irrationals for which the prime denings in a given residue class give approximation better than ℓ^{-1}.
The proof shows that Vaughan's method adapts to functions other than the unweighted von Mangoldt sum, potentially to other arithmetic functions.
The non‑Liouville result provides the first unconditional example (outside the Riemann hypothesis) of irrationals with all convergent denominators prime in a specified class.
The parameter choices (N = q², Δ ≈ C N^{-1/4} log^8 N) are sharp given the present error terms; further improvement would require stronger exponential sum estimates.

Reading between the lines

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

The error terms in Theorem 1 might be refined by using more sophisticated exponential sum techniques, possibly improving the exponent beyond 1/4.
The construction in Section 4 can likely be modified to produce irrationals with other prescribed diopantine properties, such as given irrationality measure.
The argument that S is not entirely Liouville depends on Xylouris's exponent 5.18; a better bound on the least prime would give a better uniform irrationality measure for all elements of S.

Original abstract

Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \| < \Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q> N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \| < \ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{\"o}s and Mahler from 1939.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 9,166 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.95 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \|\alpha \ell - \beta\| \ll_v \ell^{-1/4} \log^{8} \ell. $ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary. Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \| \theta \| < \Delta$ and $ 0 $ otherwise. Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q> N^{1/4}$ and $|\alpha v -a/q| \le 1/q^2$. Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation*} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation*} where $\mathcal{L}=\log (Nq/\Delta)$. This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta). \end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\| \gamma \ell \| < \ell^{-1}$. Further we prove unconditionally that not all the elements of $S$ are Liouville numbers. This addresses a question of Erd{\"o}s and Mahler from 1939.

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	Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, $\beta \in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ \Delta \in (0, 1/2)$.	background 1.00	0.05	0.01	unclear 0.87
s1	For any real $y$, let $\\|y\\|$ denote the distance from $y$ to the nearest integer.	background 0.99	0.10	0.01	asserted 0.78
s2	In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely many primes $\ell \equiv u \bmod v$ such that $ \\|\alpha \ell - \beta\\| \ll_v \ell^{-1/4} \log^{8} \ell.	result 0.94	0.96	0.72	asserted 1.00
s3	$ In order to prove this result we first prove the following general theorem and then deduce the above as a corollary.	method 0.98	0.79	0.02	asserted 0.94
s4	Before stating the result, let us define a function on $f_{\Delta}(\theta)$ on $\R$ such that $f_{\Delta}(\theta)$ is $ 1 \text{ if } \\| \theta \\| < \Delta$ and $ 0 $ otherwise.	method 0.86	0.72	0.02	asserted 0.92
s5	Further, suppose that $u, v \in \mathbb{Z}_{\ge 1}$ are coprime and $a$, $q \in \Z$ are coprime with $q> N^{1/4}$ and $\|\alpha v -a/q\| \le 1/q^2$.	background 0.33	0.27	0.02	asserted 0.67
s6	Then, for every $\epsilon \in \mathbb{R}_{>0}$ we have \begin{equation} \sum_{\substack{n=1 \\ n \equiv u \bmod v}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta) \ll_v (Nq^{-1/2} + N^{3/4} + N^{5/6}\Delta^{1/2} + (\Delta Nq)^{1/2} + N^{\epsilon}q \Delta^{1-\epsilon}) \mathcal{L}^8 \end{equation} where $\mathcal{L}=\log (Nq/\Delta)$.	result 0.94	0.92	0.10	asserted 1.00
s7	This generalises a well known result of Vaughan from 1977 proving a similar bound for the sum \begin{equation*} \sum_{\substack{n=1}}^N \Lambda(n) (f_{\Delta}(\alpha n - \beta) - 2\Delta).	background 0.67	0.37	0.04	asserted 1.00
s8	\end{equation*} In the second part of this paper, we explicitly construct an uncountable set $S \subset \R\setminus \Q$ such that for every $\gamma \in S$ there are infinitely many primes $\ell \equiv u \bmod v$ satisfying $\\| \gamma \ell \\| < \ell^{-1}$.	result 0.76	0.97	0.19	asserted 1.00
s9	Further we prove unconditionally that not all the elements of $S$ are Liouville numbers.	result 0.98	0.96	0.09	asserted 1.00
s10	This addresses a question of Erd{\"o}s and Mahler from 1939.	interpretation 0.54	0.19	0.07	asserted 0.83

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

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References

Works cited (9)

[1]

J. Borwein J, A. van der Poorten, J. Shallit, W. Zudilin, Neverending Fractions: An Introduction to Continued Fractions,Cambridge University Press, (2014)

View on Pith 2014
[2]

Y. Bugeaud, Approximation by Algebraic Numbers,Cambridge University Press, (2004)

View on Pith 2004
[3]

P . Erd¨os, K. Mahler,Some Arithmetical Properties of the Convergents of a Continued Fraction, J. London Math. Soc. 14(1), (1939), 12–18

View on Pith 1939
[4]

G. Harman,On the distribution ofαp modulo one. II., Proc. Lond. Math. Soc.3(72), (1996), 241–260

View on Pith 1996
[5]

G. Harman, Prime-Detecting Sieves (LMS-33), London Mathematical Society Monographs,Princeton University Press, (2020)

View on Pith 2020
[6]

D. R. Heath-Brown, C. Jia,The Distribution ofαpModulo One, Proc. Lond. Math. Soc.84, (2002), 79-104

View on Pith 2002
[7]

K. Matom ¨aki,The distribution ofαpmodulo one, Math. Proc. Camb. Phil. Soc.147, (2009), 267–283

View on Pith 2009
[8]

R. C. Vaughan,On the distribution ofαpmodulo 1, Mathematika24(2), (1977), 135–141

View on Pith 1977
[9]

T. Xylouris,On the least prime in an arithmetic progression and estimates for the zeros of DirichletL-functions, Acta Arith.150(1), (2011), 65–91. 19 (D. Mazumder) GITAM (DEEMED TO BE UNIVERSITY), GANDHINAGAR, RUSHIKONDA,VISHAKAPATNAM, ANDHRAPRADESH- 530045, INDIA. Email address:dwmaz1993@gmail.com (J. Sivaraman) INDIANINSTITUTE OFSCIENCEEDUCATION ANDRESE...

View on Pith 2011

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. Borwein J, A. van der Poorten, J. Shallit, W. Zudilin, Neverending Fractions: An Introduction to Continued Fractions,Cambridge University Press, (2014)

View on Pith 2014

[2] [2]

Y. Bugeaud, Approximation by Algebraic Numbers,Cambridge University Press, (2004)

View on Pith 2004

[3] [3]

P . Erd¨os, K. Mahler,Some Arithmetical Properties of the Convergents of a Continued Fraction, J. London Math. Soc. 14(1), (1939), 12–18

View on Pith 1939

[4] [4]

G. Harman,On the distribution ofαp modulo one. II., Proc. Lond. Math. Soc.3(72), (1996), 241–260

View on Pith 1996

[5] [5]

G. Harman, Prime-Detecting Sieves (LMS-33), London Mathematical Society Monographs,Princeton University Press, (2020)

View on Pith 2020

[6] [6]

D. R. Heath-Brown, C. Jia,The Distribution ofαpModulo One, Proc. Lond. Math. Soc.84, (2002), 79-104

View on Pith 2002

[7] [7]

K. Matom ¨aki,The distribution ofαpmodulo one, Math. Proc. Camb. Phil. Soc.147, (2009), 267–283

View on Pith 2009

[8] [8]

R. C. Vaughan,On the distribution ofαpmodulo 1, Mathematika24(2), (1977), 135–141

View on Pith 1977

[9] [9]

T. Xylouris,On the least prime in an arithmetic progression and estimates for the zeros of DirichletL-functions, Acta Arith.150(1), (2011), 65–91. 19 (D. Mazumder) GITAM (DEEMED TO BE UNIVERSITY), GANDHINAGAR, RUSHIKONDA,VISHAKAPATNAM, ANDHRAPRADESH- 530045, INDIA. Email address:dwmaz1993@gmail.com (J. Sivaraman) INDIANINSTITUTE OFSCIENCEEDUCATION ANDRESE...

View on Pith 2011