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arxiv: 2312.02775 · v2 · submitted 2023-12-05 · 🧮 math.NT

On the distribution of α p² modulo one in the intersection of two Piatetski--Shapiro sets

Junyi Chu , Jinjiang Li , Min Zhang This is my paper

Pith reviewed 2026-05-24 05:23 UTC · model grok-4.3

classification 🧮 math.NT
keywords Piatetski-Shapiro setsprimes in thin setsdistribution modulo oneDiophantine approximationquadratic polynomialsexponential sums
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The pith

For irrational α and any β there are infinitely many primes p in the intersection of two Piatetski-Shapiro sets satisfying ||αp² + β|| < p to the power -(14(γ1+γ2)-27)/43 + ε when 27/14 < γ1 + γ2 < 2.

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

The paper establishes a Diophantine approximation result for quadratic polynomials evaluated at primes that belong to two different Piatetski-Shapiro sets simultaneously. These sets consist of integers of the form floor(n to the power 1/γ) for fixed γ between 1/2 and 1. The proof shows that the sequence αp² + β is dense enough modulo 1 that the distance to the nearest integer drops below a positive power of 1/p for infinitely many such primes. The admissible range on the sum of the two γ parameters ensures both that the intersection contains sufficiently many primes and that the resulting exponent is positive. This extends an earlier result that treated only a single Piatetski-Shapiro set.

Core claim

Whenever α is irrational and β is real, and whenever 1/2 < γ2 < γ1 < 1 satisfy 27/14 < γ1 + γ2 < 2, there exist infinitely many primes p such that p equals both floor(n1 to the power 1/γ1) and floor(n2 to the power 1/γ2) and ||αp² + β|| is smaller than p to the power minus (14(γ1 + γ2) - 27)/43 plus an arbitrary positive ε.

What carries the argument

The intersection of two Piatetski-Shapiro sets of primes, used to produce sufficiently dense candidates for the exponential-sum estimates that control the fractional parts of αp² + β.

If this is right

  • The stated bound on ||αp² + β|| holds for every real β.
  • The same conclusion applies when the two Piatetski-Shapiro parameters satisfy the given ordering and sum constraint.
  • The result recovers and extends the single-set case previously obtained by Dimitrov.
  • The exponent on p is explicitly determined by the sum γ1 + γ2 and becomes positive precisely when that sum exceeds 27/14.

Where Pith is reading between the lines

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

  • The same method may apply to other polynomials of higher degree provided the intersection remains dense enough.
  • One could ask whether the exponent -(14(γ1+γ2)-27)/43 can be replaced by a larger (less negative) quantity while keeping the same range on γ1 + γ2.
  • The result supplies a supply of primes p for which αp² is close to an arbitrary real number modulo 1, which could be used in problems that require primes in short arithmetic progressions or in thin sets.

Load-bearing premise

The sum of the two exponents γ1 + γ2 must lie strictly between 27/14 and 2 so that the intersection is dense enough and the target exponent stays positive.

What would settle it

A concrete pair α irrational, β real, and γ1, γ2 in the stated range for which ||αp² + β|| exceeds p to the power -(14(γ1+γ2)-27)/43 for all but finitely many primes p lying in both Piatetski-Shapiro sets.

read the original abstract

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$ is an irrational number and $\beta$ is any real number, there exist infinitely many prime numbers $p$ in the intersection of two Piatetski--Shapiro sets, i.e., $p=\lfloor n_1^{1/\gamma_1}\rfloor=\lfloor n_2^{1/\gamma_2}\rfloor$, such that \begin{equation*} \|\alpha p^2+\beta\|<p^{-\frac{14(\gamma_1+\gamma_2)-27}{43}+\varepsilon}, \end{equation*} provided that $27/14<\gamma_1+\gamma_2<2$. This result constitutes an generalization upon the previous result of Dimitrov [4].

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that whenever α is irrational and β any real, there are infinitely many primes p lying in the intersection of two Piatetski-Shapiro sequences (p = ⌊n₁^{1/γ₁}⌋ = ⌊n₂^{1/γ₂}⌋ with 1/2 < γ₂ < γ₁ < 1) such that ‖α p² + β‖ < p^{-[14(γ₁ + γ₂) - 27]/43 + ε}, provided 27/14 < γ₁ + γ₂ < 2. The result is presented as a generalization of a theorem of Dimitrov.

Significance. If correct, the result extends Diophantine approximation statements for quadratic polynomials at primes to the setting of primes in intersections of two Piatetski-Shapiro sets, furnishing an explicit positive exponent in a nonempty parameter range. The explicit dependence of the exponent on γ₁ + γ₂ is a concrete strength of the stated theorem.

minor comments (3)
  1. [Abstract] Abstract, line beginning 'This result constitutes an generalization': the indefinite article should be 'a' rather than 'an'.
  2. [Abstract] Abstract: the ordering 1/2 < γ₂ < γ₁ < 1 is stated once, but the subsequent inequality uses the sum γ₁ + γ₂ without repeating the ordering; confirm that the same ordering is maintained in all statements of the main theorem and in the proof.
  3. [References] Abstract: the citation to Dimitrov [4] appears without a full bibliographic entry; ensure the reference list supplies complete details (journal, volume, year, pages) so that the claimed generalization can be compared directly with the earlier exponent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing the explicit dependence of the exponent on γ₁ + γ₂, and for recommending minor revision. No specific major comments appear in the report, so we have no point-by-point items to address. We are happy to incorporate any minor editorial changes requested by the editor.

Circularity Check

0 steps flagged

No significant circularity; generalization of external result

full rationale

The paper states a theorem extending a prior result of Dimitrov [4] on Diophantine approximation in Piatetski-Shapiro sets. The exponent -(14(γ1+γ2)-27)/43 + ε is presented as the outcome of the argument under the explicit range condition 27/14 < γ1 + γ2 < 2, which is required for positivity and density. No equations, parameters, or quantities are fitted to data within the paper itself. The central claim does not reduce by definition or construction to any input; it relies on an independent external citation rather than self-citation chains or ansatzes smuggled from the authors' own prior work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated or derivable from the given text.

pith-pipeline@v0.9.0 · 5725 in / 1137 out tokens · 20166 ms · 2026-05-24T05:23:59.490182+00:00 · methodology

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Reference graph

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