Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

Alex Iosevich; Ben Krause; F\'elix Brokering Pinilla

Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

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Pith's one-line read This paper proves that ergodic averages along integer Cantor sets converge pointwise almost everywhere for every L^p function with p≥2, in any measure-preserving system.

desk verdict A sound, genuinely new resolution of an open pointwise-convergence problem for integer Cantor sets; the proof is intricate but the core estimates check out. read the letter →

arxiv 2607.16064 v1 pith:MMMRX2T2 submitted 2026-07-17 math.DS math.CAmath.PR

Félix Brokering Pinilla , Alex Iosevich , Ben Krause This is my paper · ORCID

classification math.DSmath.CAmath.PR MSC 37A3042B2511K55

keywords integerCantorsetsergodicaveragespointwiseconvergencevariationalinequalitiesLépingle'sinequalityFouriermultiplierslacunarydifferentiationmeasures

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes pointwise convergence of ergodic averages taken along integer Cantor sets—integers whose base-$d$ expansions use only digits from a fixed proper subset $D$ containing 0—for every $L^p$ function with $p\ge 2$, in any $\sigma$-finite measure-preserving system. Earlier work had achieved only norm convergence in $L^2$, leaving the pointwise question open. The proof re-parameterizes the Cantor set via a digit-substitution bijection $\Phi$ and proves a quantitative $r$-variation estimate for the re-parameterized averages, which implies almost-everywhere convergence. The same estimate yields a Euclidean corollary: the natural Cantor measure differentiates $L^2$ functions at the self-similar scales $d^{-k}$, answering a question about lacunary differentiation.

What carries the argument

The central object is the digit-substitution bijection $\Phi:\mathbb{Z}_{\ge 0}\to C$, which writes $m$ in base $q=|D|$ and replaces each base-$q$ digit $b_j$ with $\delta_{b_j}\in D$ to form a base-$d$ integer. Its key property is the no-carry block factorization $\Phi(vq^n+r)=d^n \Phi(v)+\Phi(r)$ for $0\le v$ and $0\le r<q^n$, which yields the multiplier identity $\sigma_{u q^n}(\beta)=\sigma_u(d^n \beta)P_n(\beta)$. This identity lets the authors approximate any average over the first $M$ values of $\Phi$ by averages over a fixed mesh $G_s$, with errors controlled by the telescoping relation $|P_n|^2 Q_n = |P_n|^2 - |P_{n+1}|^2$. The remaining variation control is delegated to Lépingle's inequality, a martingale variation bound.

What would settle it

Take $D=\{0,2\}$ in base $d=3$, let $\nu$ be the natural Cantor measure on $C'$, and let $f$ be the indicator of the unit interval. Compute the sequence $I_k(x)=\int f(x-d^{-k}t)\,d\nu(t)$ for $k=0,\ldots,K$ at a fixed generic point (say $x=0.5$). Proposition 5.1 predicts that the $r$-variation of $I_k$ is bounded independent of $K$; a numerical integration showing this variation growing like $K^\alpha$ with $\alpha>0$ would refute the differentiability claim, and hence the core variational estimate.

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Extended reading notes

Core claim

The central claim is that for any $\sigma$-finite measure-preserving system $(X,\mu,T)$ and any $f\in L^p(X)$ with $2\le p<\infty$, the averages $(1/|C_N|)\sum_{n\in C_N} f(T^n x)$ converge $\mu$-almost everywhere. This is proved through a stronger variational theorem: the re-parameterized averages $A^T_M f(x) = (1/M)\sum_{m<M} f(T^{\Phi(m)}x)$ have bounded $r$-variation for every $r>2$, with $L^2$ operator norm at most a constant times $r/(r-2)$. The argument decomposes the Fourier multiplier of the averages into complete blocks via the exact factorization $\sigma_{u q^n}(\beta)=\sigma_u(d^n \beta)P_n(\beta)$, where $P_n$ is the multiplier of the $n$-th complete block. Error terms are shown to be square-summable through a telescoping identity, and Lépingle's martingale va

Load-bearing premise

The proof depends on the no-carry digit-block factorization $\Phi(vq^n+r)=d^n \Phi(v)+\Phi(r)$, which holds only because the digit set $D$ is a subset of $\{0,\ldots,d-1\}$ with $0\in D$ and $|D|\ge 2$; if the digit set allowed carries or omitted $0$, the multiplier identity would break and the entire variational decomposition would collapse.

Editorial extensions

If this is right

Pointwise convergence holds for all σ-finite measure-preserving systems, not just probability spaces, and for every f∈L^p with 2≤p<∞.
The variational estimate is quantitative: the r-variation of the re-parameterized averages is bounded in L² by a constant times r/(r−2), so the convergence is not merely qualitative.
The Euclidean corollary gives a lacunary differentiation theorem: for every f∈L²_loc(R), ∫ f(x−d^{-k}t)dν(t) → f(x) Lebesgue-a.e., where ν is the natural Cantor measure.
The maximal estimate for the re-parameterized averages transfers to a maximal bound for the original Cantor-set averages, which is what upgrades convergence from L²∩L^∞ to all L^p, p≥2.
The method also covers non-uniform Cantor measures, as noted in Remark 5.2, extending the differentiation result to a wider class of self-similar measures.

Reading between the lines

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

The no-carry block structure is the only place the specific shape of D is used; the same proof likely transfers to any digit set satisfying an analogous block separation, such as complete residue systems modulo powers of d, extending the conclusion to a broader family of self-similar integer sets.
The L²-based nature of the argument leaves the range 1≤p<2 open; if a weak-type maximal estimate could be established for the re-parameterized averages, the same blocking framework would probably yield pointwise convergence for all p>1.
One could test the quantitative strength of Proposition 5.1 numerically: for a step function f and a Cantor measure such as D={0,2}, d=3, the r-variation of the convolution sequence at a generic point should stay bounded as the number of scales grows; a power-law growth would indicate a gap in the argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is the real thing. It closes an explicitly open pointwise-convergence question for ergodic averages along integer Cantor sets, and the proof structure is sound as far as I can check. The new ingredient is the even/odd random splitting in Lemma 3.2, which is a clever way to avoid the divergent sum of Q_k while still applying Bourgain-style martingale arguments.

The paper does what a good advance should: it states the open question (from [2] and [8]), proves a quantitative r-variation estimate (Theorem 1.4) for the re-parametrized averages, transfers it via Calderon to the ergodic setting (Theorem 1.3), and then converts the parameterized averages to the original Cantor-set averages using the digit-block factorization (Lemma 4.3). The corollary on lacunary differentiation of deterministic Cantor measures is a nice payoff, and Remark 5.2 honestly flags the non-uniform extension.

I looked at the places I was most suspicious. Lemma 4.3 is solid: since D is contained in {0,...,d-1} and 0 is in D, the base-q digit blocks of v and r0 sit in disjoint positions, so the identity Phi(v q^n + r0) = d^n Phi(v) + Phi(r0) has no carry problem. The fixed-mesh approximation in Theorem 1.2 is also fine; the error is O(q^{-s}) in L-infinity, which is enough to pass to the limit. In Lemma 3.2, the even/odd splitting is legitimate: the telescoping |P_j|^2 Q_j = |P_j|^2 - |P_{j+1}|^2 is exact, and the apparent index mismatch in the even subseries is harmless because the omitted factors have modulus at most one. I did not find a circular step or a fitted parameter.

Soft spots are real but not disqualifying. The whole argument is L2-based, so the theorem stops at p >= 2; p < 2 is left open and probably needs new ideas. The variation constants carry the usual r/(r-2) factor and there is no endpoint r=2. The proof is terse in places; a referee will need to fill in routine but nontrivial details, especially in the randomization and transference steps. None of this looks like a hole.

Citation pattern is fine. The paper builds on Bourgain, Lepingle, and Calderon, and cites the relevant Cantor-set literature. I do not see the main theorem depending on self-citation.

Who gets value: ergodic theorists and harmonic analysts working on pointwise convergence, lacunary differentiation, and variational estimates. I would send it to a serious referee; it deserves the full review process.

Referee Report

0 major / 4 minor

Summary. The paper proves pointwise almost-everywhere convergence of ergodic averages along integer Cantor sets C = {Σ a_j d^j : a_j ∈ D} for functions in L^p, 2 ≤ p < ∞, in any σ-finite measure-preserving system. The proof proceeds by re-parameterizing the averages via the digit-substitution map Φ, establishing an ℓ²-valued r-variation estimate for the re-parameterized sequence-space operators A_M (Theorem 1.4), transferring this estimate to measure-preserving systems via Calderón's transference principle (Theorem 1.3), and then using a fixed-mesh approximation by times of the form u q^n to upgrade variation convergence to full convergence of the original averages (Theorem 1.2). A Euclidean corollary gives Lebesgue-a.e. differentiation of convolutions with self-similar Cantor measures at scales d^{-k}.

Significance. If the proof is correct, this resolves an open pointwise-convergence question for integer Cantor sets, extending the L² convergence of [2] and Bourgain's earlier maximal estimate. The main strength is the quantitative r-variation estimate with explicit constant r/(r-2) and no fitted parameters. The proof is self-contained, builds on standard tools (Lépingle's inequality, Calderón's transference, Plancherel), and the structural Lemma 4.3 is clean and correctly handles the no-carry digit-block factorization. The Euclidean differentiation corollary is a natural and attractive application. Overall this is a substantial contribution to the ergodic theory of sparse sets.

minor comments (4)

[§4, Proposition 4.1] In the treatment of the endpoint term P_J H_J, the phrase 'dominating the variation by a square sum' is terse. The bound V_r(a_n) ≤ 2(Σ |a_n|^2)^{1/2} justifies this step, but it would help readers if this inequality were stated explicitly, since at first glance ℓ² summability of the sequence values does not obviously control r-variation.
[§3, Lemma 3.2] In the even/odd subseries estimates, the displayed bound for E|c_j|² omits a factor of modulus ≤ 1 (e.g., |m_{2j}|² in the odd case and |m_{2j-1}|² in the even case). The resulting telescoping is valid, but the omission may look like an indexing error; a parenthetical clarification would improve readability.
[§2.3 and §4, Theorem 1.2] The relation C_N = Φ({1,...,M(N)}) is used to pass from convergence of the re-parameterized averages to convergence of C_N averages. Since C_N is defined as C ∩ {1,...,N}, excluding 0, it would be helpful to state explicitly that Φ(0)=0 and Φ is strictly increasing, so the first M(N) positive elements of C are exactly C_N.
[§4, Theorem 1.2] The extension from L² ∩ L^∞ to L^p is compressed into the phrase 'we extend this to all L^p(X) functions...'. A brief sentence explaining the use of the maximal estimate (Corollary 4.2 transferred) and interpolation would make the argument fully transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained given Lépingle's inequality and Calderón's transference principle.

full rationale

The paper's central claim, Theorem 1.2, is established by an explicit chain of estimates: Lemma 3.1 derives a variational bound for random Fourier multipliers from Lépingle's inequality and orthogonality in L2(Ω); Lemma 3.2 applies this to the Cantor-type multipliers via an even/odd splitting; Proposition 4.1 then proves uniform variation bounds for the complete-block averages B_n, with the auxiliary functions R and G chosen only to satisfy the pointwise estimate |R(t)|+|G(t)|≲Q(t). No fitted parameter is used, and no quantity called a prediction is instead obtained from data. The transition from variation bounds to a.e. convergence (Corollary 4.2, Theorem 1.4, and the proof of Theorem 1.2) is a standard fixed-mesh approximation argument: Lemma 4.3 supplies the exact factorization σ_{u q^n}(β)=σ_u(d^n β)P_n(β), and the error terms are controlled by a telescoping square-sum. The paper invokes Calderón's transference principle and Lépingle's inequality as external results; these are not supplied by the authors themselves and are not used circularly. The few self-references or related papers (e.g., [9], a survey by Krause, and [2], prior work on L2 convergence) are contextual or motivational and are not load-bearing for Theorem 1.2: the proof does not reduce to those citations but rather re-proves the needed quantitative bounds directly. There are no instances of an ansatz smuggled in by self-citation, no self-definitional identities, and no fitted input renamed as a prediction. The corollary on lacunary differentiation, Corollary 1.5, is obtained from Proposition 5.1 using a direct rescaling and a standard localization argument; it does not assume the conclusion. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof relies on classical external theorems (Lépingle, Calderón, Plancherel) and on structural properties of the digit set. No free parameters are fitted; auxiliary functions R,G are explicit and carry no independent existence claim.

assumptions (4)

standard math Lépingle's inequality (Proposition 2.1): ‖V^r(E(F|B_j))‖_L² ≲ r/(r−2)‖F‖_L² for any monotone filtration.
Used as the principal martingale input in Lemma 3.1 and Proposition 4.1; Section 2.4.
standard math Calderón's transference principle (Theorem 1.3 from 1.4).
Used in Section 4 to transfer ℓ²(Z) sequence-space estimates to arbitrary σ-finite measure-preserving systems.
standard math Fourier inversion and Plancherel on ℓ²(Z) and L²(R).
Used throughout Sections 2–5 to move between functions and their Fourier transforms.
domain assumption The fundamental multiplier m(t)=q^{-1}∑_{a∈D}e(−at) satisfies |m(t)|≤1 with equality only at t∈Z, and Q(t)=1−|m(t)|²≈dist(t,Z)².
This is a property of the finite digit set D; it underlies (2.4)–(2.7) and the telescoping estimates. It is derived from the hypotheses on D, not imported from outside.

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Pith. "Pith review of Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets." pith.science (2026). https://pith.science/paper/MMMRX2T2

@misc{pith2026260716064,
  author       = {Pith},
  title        = {Pith review of: Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMMRX2T2}},
  note         = {Machine review of arXiv:2607.16064}
}

abstract

Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ d\nu(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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Burgin; A

A. Burgin; A. Fragkos; M. Lacey; D. Mena; M. Carmen Reguera.Integer Cantor Sets: Arithmetic Combinatorial Properties. Preprint,https://arxiv.org/abs/2602.15292. (Pages 1 and 2)

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[2] [8]

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[3] [1]

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[4] [3]

Burgin; A

A. Burgin; A. Fragkos; M. Lacey; D. Mena; M. Carmen Reguera.Szemer´ edi’s Theorem Along Cantor Sets of Integers. Preprint,https://arxiv.org/abs/2602.15299. (Page 2)

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[5] [4]

Calder´ on.Ergodic theory and translation invariant operators.Proc

A. Calder´ on.Ergodic theory and translation invariant operators.Proc. Nat. Acad. Sci., USA 59 (1968), 349-353. (Page 3)

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[6] [5]

N. J. Fine.The distribution of the sum of digits (mod p). Bull. Amer. Math. Soc.71(1965), 651–652. (Page 1) 16Pointwise Convergence Along Integer Cantor Sets

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[7] [6]

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B. Green.Waring’s problem with restricted digits.Compos. Math. 161 (2025), no. 2, 341–364. (Pages 1 and 4)

2025

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Jones; A

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2008