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arxiv: 2605.18676 · v1 · pith:W2OUGUFOnew · submitted 2026-05-18 · 🧮 math.NT · math.CO

Linear equations in Piatetski-Shapiro primes

Xuancheng Shao , Yu-Chen Sun This is my paper

Pith reviewed 2026-05-20 08:19 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Piatetski-Shapiro primesarithmetic progressionsnilsequencesdiscorrelation estimateslinear equationsasymptotic formulasnumber theory
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The pith

Piatetski-Shapiro primes contain infinitely many k-term arithmetic progressions when gamma exceeds 1 minus 2 to the minus Ck.

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

The paper proves discorrelation estimates between the Piatetski-Shapiro prime set and arbitrary nilsequences provided gamma lies sufficiently close to 1. These estimates yield an asymptotic formula for the number of solutions to any finite-complexity linear system inside the set, including the count of k-term arithmetic progressions up to any large N. In particular the work shows an absolute constant C exists so that whenever gamma satisfies 1 minus 2 to the power of minus Ck less than gamma less than 1 the set contains infinitely many nontrivial k-term progressions. This range improves earlier triple-exponential thresholds. The result matters because it demonstrates that these primes, although thinner than ordinary primes, still support the same linear configurations that ordinary primes are known to contain.

Core claim

We establish discorrelation estimates between the Piatetski-Shapiro prime set P_gamma and arbitrary nilsequences when gamma in (0,1) is sufficiently close to 1. As an application, we establish an asymptotic formula for the number of solutions in P_gamma to any finite-complexity system of linear equations, including for the number of k-term arithmetic progressions in P_gamma up to a threshold N for any given k greater than or equal to 3. Furthermore, we show that there exists an absolute constant C greater than 0 such that if 1 minus 2 to the minus Ck less than gamma less than 1, then the Piatetski-Shapiro primes P_gamma contain infinitely many non-trivial k-term arithmetic progressions. This

What carries the argument

Discorrelation estimates between the Piatetski-Shapiro prime set P_gamma and arbitrary nilsequences for gamma close to 1, which transfer uniformity properties from ordinary primes to this sparser set.

If this is right

  • The number of solutions to any finite-complexity linear system inside P_gamma admits an asymptotic formula.
  • The number of k-term arithmetic progressions inside P_gamma up to N is given by the expected main term for every fixed k at least 3.
  • There exist infinitely many nontrivial k-term arithmetic progressions inside P_gamma whenever gamma lies in the interval 1 minus 2 to the minus Ck less than gamma less than 1.

Where Pith is reading between the lines

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

  • The same discorrelation technique could apply to other sparse subsets of primes whose distribution is controlled at similar scales.
  • The range on gamma might be pushed still closer to 1 by refining the nilsequence correlation bounds further.
  • These linear configurations inside P_gamma suggest the possibility of similar results for polynomial systems once suitable higher-degree discorrelation is established.

Load-bearing premise

The new discorrelation estimates hold for gamma sufficiently close to 1, in particular inside the interval 1 minus 2 to the minus Ck less than gamma less than 1.

What would settle it

An explicit computation or sieve that shows the count of k-term arithmetic progressions inside P_gamma up to some large N falls outside the predicted main term for a value of gamma inside the claimed range.

read the original abstract

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_{\gamma} := \{p \text{ is prime and } p = \lfloor n^{1/\gamma} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when $\gamma \in (0,1)$ is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in $\mathcal{P}_{\gamma}$ to any "finite-complexity" system of linear equations, including for the number of $k$-term arithmetic progressions in $\mathcal{P}_{\gamma}$ up to a threshold $N$ for any given $k \geq 3$. Furthermore, we show that there exists an absolute constant $C>0$ such that if \[ 1 - 2^{-Ck} < \gamma < 1, \] then the Piatetski-Shapiro primes $\mathcal{P}_{\gamma}$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $\gamma$ obtained by Li and Pan, which is of triple exponential type.

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

1 major / 2 minor

Summary. The manuscript establishes discorrelation estimates between the Piatetski-Shapiro prime set P_gamma and arbitrary nilsequences for gamma in (0,1) sufficiently close to 1, extending prior results on linear and polynomial phases. As an application, it derives asymptotic formulas for the number of solutions in P_gamma to any finite-complexity system of linear equations (including k-term APs for k >= 3) and proves that there exists C > 0 such that gamma > 1 - 2^{-C k} implies infinitely many non-trivial k-term APs in P_gamma, improving on the triple-exponential range of Li-Pan.

Significance. If the discorrelation bounds hold with the stated quantitative control, the work would provide a meaningful advance in the arithmetic structure of thin prime sets, yielding a single-exponential dependence on k for k-APs that is substantially stronger than previous results. The extension to arbitrary nilsequences is a natural and useful strengthening of earlier discorrelation theorems.

major comments (1)
  1. [§3 (discorrelation estimates) and the k-AP application in §5] The main discorrelation theorem (likely Theorem 1.2 or the statement in §3) asserts bounds sufficient for the range 1 - 2^{-C k} < gamma < 1 in the k-AP application. However, the nilsequences arising from the Gowers-uniformity decomposition or linear-equation transference have dimension, step, and Lipschitz constants that grow with k. The error term must be shown to depend on these parameters in a manner (at worst single-exponential in a controlled way) that preserves the claimed single-exponential range for gamma; explicit tracking of this dependence through the proof is required to confirm the improvement over Li-Pan is not lost.
minor comments (2)
  1. [Introduction] The notion of 'finite-complexity' linear systems is used repeatedly but should be defined or referenced explicitly in the introduction for readers unfamiliar with the Green-Tao-Ziegler framework.
  2. [§2 (preliminaries)] Notation for the nilsequence parameters (e.g., the Lipschitz constant and the dimension of the nilmanifold) should be introduced consistently before their first use in the estimates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion to make the parameter dependence explicit. We address the major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3 (discorrelation estimates) and the k-AP application in §5] The main discorrelation theorem (likely Theorem 1.2 or the statement in §3) asserts bounds sufficient for the range 1 - 2^{-C k} < gamma < 1 in the k-AP application. However, the nilsequences arising from the Gowers-uniformity decomposition or linear-equation transference have dimension, step, and Lipschitz constants that grow with k. The error term must be shown to depend on these parameters in a manner (at worst single-exponential in a controlled way) that preserves the claimed single-exponential range for gamma; explicit tracking of this dependence through the proof is required to confirm the improvement over Li-Pan is not lost.

    Authors: We agree that explicit tracking strengthens the presentation. In the proof of the discorrelation theorem in Section 3, the error bounds are obtained via the quantitative inverse theorem for Gowers norms (with dependence single-exponential in the dimension and step) combined with the transference principle for the linear equations. For the k-AP application in Section 5, the nilsequences arising in the Gowers-uniformity decomposition have dimension and step at most exponential in k, and the Lipschitz constants are controlled polynomially in these quantities. Consequently the overall error remains single-exponential in k, which is compatible with the claimed range gamma > 1 - 2^{-C k}. We will add a dedicated paragraph at the end of Section 3 and a short subsection in Section 5 that records these dependencies explicitly, together with the resulting bound on the constant C. revision: yes

Circularity Check

0 steps flagged

No circularity: new discorrelation estimates are independently derived and applied

full rationale

The paper proves fresh discorrelation bounds between the Piatetski-Shapiro set P_gamma and arbitrary nilsequences when gamma is sufficiently close to 1, extending earlier results on linear and polynomial phases via its own analytic arguments. These bounds are then fed into Gowers-uniformity or transference machinery to obtain the asymptotic for finite-complexity linear systems and the k-AP count. The claimed single-exponential range 1-2^{-Ck} is asserted to follow from quantitative control of nilsequence parameters inside the new estimates; no equation or step reduces by construction to a fitted parameter, a self-citation chain, or a renaming of prior inputs. Citations to Li-Pan supply only the baseline triple-exponential range and are not load-bearing for the improvement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard tools from higher-order Fourier analysis without introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard results on nilsequences and discorrelation estimates from prior works in additive combinatorics
    Invoked to establish the new discorrelation for P_gamma when gamma near 1.

pith-pipeline@v0.9.0 · 5751 in / 1360 out tokens · 55019 ms · 2026-05-20T08:19:28.944845+00:00 · methodology

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Reference graph

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