arXiv:2608.03767v1

Sharp Orlicz Endpoints for Spatial-Temporal Ergodic Averaging

Jie Li

math.DS37A3042B2546E30

Paper metadata

Original submission: August 4, 2026
Claimed MSC: 37A30, 42B25, 46E30
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 64 pages

Abstract

We study the composition of temporal ergodic averaging with spatial averaging over shrinking metric balls, and determine sharp Orlicz endpoints for the corresponding unrestricted joint limit. For consecutive Birkhoff averages normalized by $NΛ_q(N)$ ( $q\ge 0$ ), the sharp Orlicz endpoint is $L\log_{q+1}L$ . In particular, the ordinary case $q=0$ yields an $L\log L$ local joint convergence theorem under the Lebesgue differentiation property alone, while $L^1$ fails even on the Euclidean interval, answering two questions of Young. The same $L\log_{q+1}L$ endpoint holds for prime averages for every $q\ge 1$ . For arbitrary time sequences, $L\log_qL$ always suffices at the same normalization $NΛ_q(N)$ ( $q\ge 1$ ), and this endpoint is sharp in the Orlicz sense for fixed-base exponential sequences $k^n$ ( $k\ge 2$ ) and for sequences with polynomial ratio separation, including $n!$ . The positive results rest on a local stability principle: under the ball Lebesgue differentiation property, pointwise temporal convergence lifts to the local joint limit whenever the associated temporal maximal function admits an $L^1$ majorant. The additional logarithm in the regular-time case comes from lifting restricted logarithmic maximal estimates from sets to general functions. The arbitrary-sequence theorem uses a dyadic decomposition instead. The lower bounds are local $\infty$ -sweeping out constructions, obtained by weighted local sweeping out for polynomial-growth regular times, by residue constructions for polynomially ratio-separated sequences, and by digit constructions for fixed-base exponentials.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The claimed sharp local-observation endpoints for consecutive and prime times, the universal positive endpoint for arbitrary time sequences, and the sharpness results for polynomially ratio-separated and fixed-base exponential sequences are correct as stated.

Theorem A(i)Correct

Sharp endpoint for consecutive times

Pages 4 and 29 · Theorem A(i) and its proof · arXiv:2608.03767v1

For $a_n=n-1$ , the maximal ergodic theorem supplies the restricted maximal estimate used by the logarithmic lifting argument. This gives an $L^1$ majorant for the averages normalized by $N\Lambda_q(N)$ on $L\log_{q+1}L$ . Birkhoff's theorem gives the limit $\mathbb E(f\mid\mathcal I)$ when $q=0$ , while the extra factor $\Lambda_q(N)\to\infty$ gives limit $0$ for $q\geq1$ . The polynomial-growth sweeping-out construction proves failure in every Orlicz class with $\Phi(t)=o(tL_{q+1}(t))$ , establishing the asserted sharp endpoint.

Full paper, version 1 ↗

Theorem A(ii)Correct

Sharp endpoint for prime times

Pages 4 and 30–34 · Theorem A(ii) and its proof · arXiv:2608.03767v1

Trojan's restricted prime maximal estimate gives $\mu\{P^*\mathbf1_E>\alpha\}\lesssim \frac{\mu(E)}{\alpha}\log^2\!\left(\frac e\alpha\right).$ Layer-cake integration yields the cubic logarithmic set estimate in Proposition 3.14, and Proposition 3.16 converts it into every restricted estimate $\mathrm{RME}_q$ . The general lifting and local-stability theorems therefore give convergence to $0$ on $L\log_{q+1}L$ . The elementary bound $p_n\ll n^2$ places the primes within the polynomial-growth lower theorem, so the endpoint is sharp.

Trojan, Endpoint estimates for the maximal function over prime numbers ↗

Theorem B(i)Correct

Universal positive endpoint

Pages 4–5 and 37–40 · Theorem B(i) and its proof · arXiv:2608.03767v1

For an arbitrary nonnegative integer sequence $(a_n)$ , dyadic truncation and measure preservation give an $L^1$ bound for the positive maximal function $M_{a,q}f=\sup_{N\geq2}\frac1{N\Lambda_q(N)}\sum_{n=1}^N |f|\circ T^{a_n}$ on $L\log_qL$ . Bounded truncations converge uniformly to $0$ , and the maximal estimate makes the truncation error vanish in measure and then almost everywhere. The same integrable majorant activates the local-stability theorem on every underlying metric probability space with the Lebesgue differentiation property.

Full paper, version 1 ↗

Theorem B(ii)Correct

Sharp sparse-sequence endpoints

Pages 5 and 45–64 · Theorem B(ii), Propositions 4.13 and 4.18 · arXiv:2608.03767v1

Polynomial ratio separation allows disjoint residue blocks whose normalized orbit sums are large on sets of uniformly positive measure, while the base- $k$ digit construction supplies the analogous finite stages for $a_n=k^n$ . In both cases the stage parameters can be chosen so that the Orlicz cost is summable whenever $\Phi(t)=o(tL_q(t))$ . The interval assembly theorem then produces local $\infty$ -sweeping-out counterexamples, which combine with Theorem B(i) to give the claimed sharp $L\log_qL$ endpoint.

02Proofs5 reported findingsCorrect

The central proofs are correct and complete. The local-stability reduction, logarithmic restricted-to-strong lifting, universal dyadic maximal estimate, weighted interval assembly, and the two sparse finite-stage constructions all supply the hypotheses used in the main theorems.

Theorem 2.12Correct and complete

Integrable domination implies local stability

Pages 15–17 · Theorem 2.12 · arXiv:2608.03767v1

Egorov sets $K_j$ are chosen with summable complements, so Borel–Cantelli gives eventual membership almost everywhere. At a point that is also a Lebesgue point of every $H\mathbf1_{K_j^c}$ , uniform convergence on one eventual $K_j$ bounds the tail by $\eta/2+H\mathbf1_{K_j^c}$ . Averaging and taking $r\downarrow0$ removes the second term. The order of the limits is the one required by the definition of local stability, and no uniform differentiation assertion beyond countably many integrable functions is used.

Propositions 3.2 and 3.16Correct and complete

Restricted logarithmic estimates lift to an integrable maximal majorant

Pages 21–23 and 31–33 · Propositions 3.2 and 3.16 · arXiv:2608.03767v1

The layer-cake representation reduces a nonnegative function to indicator functions and Tonelli's theorem permits integration of the restricted estimates. The rearrangement bound contributes exactly one iterated logarithm. For the prime application, splitting at $m_R=\lceil\exp(1+L_1(R)^A)\rceil$ , with $R=\mu(E)^{-1}$ , controls short times by summing $1/(n\Lambda_q(n))$ and long times by the ordinary maximal function divided by $\Lambda_q(m_R)$ . Lemma 3.15 gives both required comparisons, so all constants remain uniform in the system.

Theorem 4.3Correct and complete

Dyadic maximal estimate for arbitrary time sequences

Pages 37–40 · Lemmas 4.1–4.2 and Theorem 4.3 · arXiv:2608.03767v1

At dyadic level $t$ , the weight $\omega_q(t)=1/(t\Lambda_{q-1}(t))$ permits the decomposition of $|f|$ at height $\omega_q(t)^{-1}$ . The bounded part contributes at most one and Tonelli plus invariance controls the tails by $|f|\sum_{t<|f|}\omega_q(t)\lesssim |f|(1+L_q(|f|))$ . Monotonicity and comparability of $\Lambda_q(2^t)$ transfer the estimate to every $N$ . Luxemburg homogeneity and truncation then prove both the maximal bound and almost-everywhere convergence.

Theorem 4.6 and Corollary 4.7Correct and complete

Finite stages assemble into local sweeping out

Pages 45–51 · Theorem 4.6 and Corollary 4.7 · arXiv:2608.03767v1

Each finite cyclic witness is placed on sufficiently fine interval towers and transferred by a measure-preserving interval permutation. The chosen exceptional measures are summable, while the target hit sets have a fixed positive lower measure at every stage. Borel–Cantelli and the density construction therefore force arbitrarily large local normalized averages almost everywhere. The summed Orlicz costs ensure that the assembled nonnegative function belongs to $L^\Phi$ , so the construction proves exactly the lower assertion used in the endpoint definition.

Propositions 4.13 and 4.18Correct and complete

Residue and digit constructions supply admissible sparse stages

Pages 53–64 · Propositions 4.13 and 4.18 · arXiv:2608.03767v1

For polynomial ratio separation, successive blocks can be made disjoint modulo a large prime and assigned enough residues to give the required hit count while their density remains within the prescribed budget. For $k^n$ , occurrences of primitive length- $n_0$ base- $k$ words yield $R_t$ visits to the corresponding cylinder; distinct word classes give disjoint cylinders, and primitive words occupy at least half of all words once $n_0$ is large. The stage choices $\eta_j\asymp j^{-2}$ and rapidly increasing amplitudes make the hit levels diverge while convexity and $\Phi(t)=o(tL_q(t))$ make the Orlicz-cost series converge.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.03767v1
Authors listed: Jie Li
Audit date: August 18, 2026

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Sharp Orlicz Endpoints for Spatial-Temporal Ergodic Averaging

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Sharp endpoint for consecutive times

Sharp endpoint for prime times

Universal positive endpoint

Sharp sparse-sequence endpoints

Integrable domination implies local stability

Restricted logarithmic estimates lift to an integrable maximal majorant

Dyadic maximal estimate for arbitrary time sequences

Finite stages assemble into local sweeping out

Residue and digit constructions supply admissible sparse stages

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