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arxiv: 1907.04753 · v1 · pith:6PYAAOG4new · submitted 2019-07-10 · 🧮 math.DS · math.CA· math.NT

Endpoint estimates for the maximal function over prime numbers

Bartosz Trojan This is my paper

Pith reviewed 2026-05-24 23:21 UTC · model grok-4.3

classification 🧮 math.DS math.CAmath.NT
keywords ergodic averagesprimesOrlicz spacespointwise convergencemaximal functionsdynamical systemsmeasure theory
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The pith

In ergodic dynamical systems, averages along primes converge almost everywhere for functions in L(log L)^2(log log L).

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

The paper establishes pointwise convergence of ergodic averages taken along the sequence of primes. For any ergodic transformation T, if f is integrable enough to belong to the Orlicz space L(log L)^2(log log L), then the averages 1/π(N) sum_{p prime <=N} f(T^p x) converge for almost every x. This result comes from bounding the maximal operator associated to these averages in the corresponding Orlicz norm. The endpoint nature of the space means the result is sharp in terms of integrability requirements.

Core claim

Given an ergodic dynamical system (X, B, μ, T), the ergodic averages along primes converge μ-almost everywhere for every f in the Orlicz space L(log L)^2(log log L)(X, μ).

What carries the argument

The maximal function over primes, defined as the supremum over N of the absolute value of the prime average of f composed with T^p, whose weak-type estimates in the Orlicz space imply the a.e. convergence.

If this is right

  • The pointwise ergodic theorem holds along the primes under this integrability condition.
  • The result applies to all ergodic measure-preserving transformations.
  • Convergence holds without additional assumptions on the system beyond ergodicity.
  • The Orlicz space condition is sufficient for the maximal inequality to hold.

Where Pith is reading between the lines

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

  • This may allow extension of similar convergence results to other lacunary sequences of integers.
  • Similar endpoint estimates could be pursued for averages along other number-theoretic sets like squares or polynomials.
  • The technique might connect to questions in harmonic analysis on groups or other dynamical settings.

Load-bearing premise

The dynamical system is ergodic.

What would settle it

Exhibiting an ergodic system and a function in L(log L)^2(log log L) for which the prime averages fail to converge on a positive measure set.

read the original abstract

Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, the ergodic averages \[ \frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p x\big), \] converge for $\mu$-almost all $x \in X$, where $\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\pi(N) = \# \mathbb{P}_N$.

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 / 1 minor

Summary. The paper proves that given an ergodic dynamical system (X, B, μ, T), the ergodic averages (1/π(N)) ∑_{p ∈ P_N} f(T^p x) converge μ-almost everywhere for every f in the Orlicz space L(log L)^2(log log L)(X, μ).

Significance. If the estimates hold, the result would be significant for ergodic theory: it reaches the endpoint in the Orlicz scale for pointwise convergence of prime averages, extending the classical Birkhoff theorem along a thin set of integers while controlling the maximal operator at the critical integrability threshold.

minor comments (1)
  1. [Abstract] The provided text consists only of the abstract; without the body of the manuscript the details of the maximal inequality, the approximation argument, and the error control cannot be verified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our result on a.e. convergence of prime ergodic averages at the Orlicz endpoint L(log L)^2(log log L). We appreciate the recognition of its potential significance for extending Birkhoff's theorem along primes. No major comments were provided in the report, so we have no point-by-point responses to offer. The manuscript stands as submitted.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a pointwise convergence theorem for prime-number ergodic averages via a maximal inequality at the Orlicz endpoint L(log L)^2(log log L) followed by a standard density argument on a dense subclass. This structure is a direct analytic proof relying on external tools from harmonic analysis and ergodic theory (e.g., transference principles and known prime-number estimates) without any reduction of the target statement to a fitted parameter, self-definition, or load-bearing self-citation chain. The ergodicity hypothesis is stated explicitly as an input rather than derived internally. The derivation remains self-contained and does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; the result rests on standard ergodic-theory assumptions and prime-number counting.

axioms (2)
  • domain assumption The dynamical system (X, B, μ, T) is ergodic and measure-preserving
    Explicitly stated as given in the abstract.
  • standard math π(N) denotes the prime-counting function with the usual asymptotic properties
    Used to normalize the averages over P_N.

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discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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    I/n.sc/t.sc/r.sc/o.sc/d.sc/u.sc/c.sc/t.sc/i.sc/o.sc/n.sc Let ( X, B, µ, T) be an ergodic dynamical system, that is ( X, B, µ) is a probability space with a measurable and measure preserving transformation T : X → X. The classical Birkhoff theorem [ 2] states that for any function f from L p( X, µ) with p ∈ [1, ∞) , the ergodic averages 1 N N − 1/summationd...

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    G/a.sc/u.sc/s.sc/s.sc /s.sc/u.sc/m.sc/s.sc We start by recalling some basic facts from number theory. A g eneral reference here is the book [ 17]. A homomorphism χ: (Z/ qZ) × → C×, is called a Dirichlet character modulo q. The simplest example, called the principal character modulo q, is defined as 1q( x) = { 1 if gcd( x, q) = 1, 0 otherwise. A character χ...

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    A/p.sc/p.sc/r.sc/o.sc/x.sc/i.sc/m.sc/a.sc/t.sc/i.sc/n.sc/g.sc /m.sc/u.sc/l.sc/t.sc/i.sc/p.sc/l.sc/i.sc/e.sc/r.sc/s.sc Let us denote by A N the averaging operator over prime numbers, that is for a func tion f : Z → C we have A N f ( x) = 1 π( N) /summationdisplay.1 p ∈P N f ( x + p) where PN = [1, N] ∩ P and π( N) = #PN . Since sums over primes are very ir...

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    Before embarking on the proof, let us recall two lemmas essential for the argument

    E/q.sc/u.sc/i.sc/d.sc/i.sc/s.sc/t.sc/r.sc/i.sc/b.sc/u.sc/t.sc/i.sc/o.sc/n.sc /o.sc/f.sc /w.sc/e.sc/a.sc/k.scℓ1 /n.sc/o.sc/r.sc/m.sc/s.sc In this section we prove that the maximal function associate d with kernels ( M β 2n : n ∈ N0) has weak ℓ1( Z) - norm equidistributed in residue classes. Before embarking on the proof, let us recall two lemmas essential ...

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    imply that QJr ( λ) ≤ J1( λ/ 2) + . . . + JQ( λ/ 2) + Cλ− 1Q22− 4s/bardblex /bardblexF − 1 (ηs ˆf ) /bardblex /bardblex ℓ1 ≲ λ− 1 ( 1 + Q22− 4s ) /bardblex /bardblexF − 1 (ηs ˆf ) /bardblex /bardblex ℓ1 ≲ λ− 1/bardblex /bardblexF − 1 (ηs ˆf ) /bardblex /bardblex ℓ1, where the last inequality is a consequence of 1 ≤ Q ≤ 22s. Therefore, in view of Lemma 4.2...

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    Theorem 5.1

    ℓ2 /t.sc/h.sc/e.sc/o.sc/r.sc/y.sc We are now in the position to prove ℓ2( Z) boundedness of the maximal function associated to the multi pliers ( νs n : n ∈ N) . Theorem 5.1. For eachǫ >0 there is C > 0 such that for all s ∈ N0, and any finitely supported function f : Z → C,/bardblex /bardblex /bardblex sup n∈N /barex /barex F − 1 (νs n ˆf ) /barex /barex ...

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    imply (21). It remains now to treat supremum over n ≥ 2s+4. For each 1 2 ≤ β <1 we set R β s = { a/ q ∈ Rs : βq = β } . and R1 s = Rs. In view of the Landau’s theorem [ 17, Corollary 11.9], there are O( log s) distinct β’s. Therefore, it suffices to show the following claim. Claim 5.3. For each ǫ >0 there is C > 0 such that for all s ∈ N0, 1 2 ≤ β≤ 1, any fi...

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    Therefore, Qs/summationdisplay.1 u=1 /bardblex /bardblexI( x, x + u) /bardblex /bardblex2 ℓ2( x) ≲ 2− s( 1− ǫ) Qs ∥ f ∥2 ℓ2, which together with (

    we get /bardblex /bardblexJ( x, x + u) /bardblex /bardblex2 ℓ2( x) = /uni222B.dsp1 0 /barex /barex /barex /summationdisplay.1 a/ q ∈R β s G( χq, a) e2π iξ ua/ q ηs( ξ− a/ q) /barex /barex /barex 2 | ˆf ( ξ)| 2 dξ ≲ 2− s( 1− ǫ) ∥ f ∥2 ℓ2. Therefore, Qs/summationdisplay.1 u=1 /bardblex /bardblexI( x, x + u) /bardblex /bardblex2 ℓ2( x) ≲ 2− s( 1− ǫ) Qs ∥ f ∥...

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    imply (23) and the theorem follows. □ Given t > 0 and n > t, we define the multiplier Πt n( ξ) = /summationdisplay.1 0≤ s ≤ √ t νs n( ξ) = /summationdisplay.1 0≤ s ≤ √ t /summationdisplay.1 a/ q ∈Rs ˆL a, q 2n ( ξ− a/ q) ηs( ξ− a/ q) . Corollary 5.4. There are C, c > 0 such that for each t > 0, and any finitely supported function f ∈ Z → C,/bardblex /bardbl...

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    A/p.sc/p.sc/l.sc/i.sc/c.sc/a.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section we show two applications of Theorem 6.3 and Corollary 6.5. First, we prove that the restricted weak Orlicz estimates together with strong ℓ2 bounds are sufficient to get ℓp maximal inequalities for all 1 < p ≤ 2. Next, we conclude almost everywhere convergence of ergodic averages for f...

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    we have /bardblex /bardblex f /bardblex /bardblex L( log L) 2( log log L) = /uni222B.dsp1 0 f ∗( t) φ(t− 1) dt ≥ /summationdisplay.1 j ≥ 1 aj φ( 2j ) 2− j− 1 ≥ 1 8 /summationdisplay.1 j ≥ 1 aj µ( Aj) log2 ( e µ( Aj) ) log( j + 1) , which together with ( 34) conclude the proof. □ R/e.sc/f.sc/e.sc/r.sc/e.sc/n.sc/c.sc/e.sc/s.sc

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