Sufficiency of Bourgain's bounded-entropy criterion for pointwise convergence of averages of commuting positive isometries

Bourgain's paper on pointwise ergodic theorems for arithmetic sets (Publications Mathématiques de l'IHÉS 69, 1989) closes with a general necessary condition for almost-everywhere convergence: if the averages $A_N f=(1/N)\sum_{n\le N}T_n f$ of commuting positive isometries $T_n$ of $L^2(\mu)$ converge almost everywhere, then for each $\delta>0$ the $\delta$-metrical entropy of the orbit $\{A_N f\}$ is bounded, uniformly over the $L^2$ unit ball (proved in his 1988 paper on almost sure convergence and bounded entropy). Whether this entropy condition is sufficient — whether bounded orbit entropy forces convergence — was left open and remains open; no general criterion for commuting positive isometries of an infinite-dimensional $L^2(\mu)$ is known. The paper's other open directions have since been resolved by later work: the $L^1$ theory for squares and primes was settled negatively by Buczolich and Mauldin (2010), who proved the squares are $L^1$-universally bad (in every ergodic system some $L^1$ function's averages along the squares diverge), extended to $d$-th powers and the primes by LaVictoire (2011); the $L^p$ theory for averages along $\lfloor P(n)\rfloor$ for real polynomials $P$ was completed by Krause (2026), with an appendix by Wierdl reaching the full expected range; and the $L^r$, $r>1$, primes case predates the paper itself (Wierdl 1988). The unresolved target is the general necessary-and-sufficient criterion: since necessity is already known, a positive answer to the sufficiency question supplies such a criterion directly, whereas a negative answer is realised by an explicit separation satisfying the uniform entropy bound.

Background

Let $(X,\mathcal B,\mu)$ be a probability space equipped with a measure-preserving transformation $T$ (a map with $\mu(T^{-1}A)=\mu(A)$ for every measurable set $A$). Birkhoff's pointwise ergodic theorem states that the ergodic averages $(1/N)\sum_{n\le N}f(T^n x)$ converge for almost every $x$ whenever $f\in L^1(\mu)$; when the system is ergodic, meaning that every measurable set $A$ with $T^{-1}A=A$ has measure $0$ or $1$, the limit is the space average $\int f\,d\mu$. A central program, to which the source paper belongs, asks how far this survives when the exponent $n$ is restricted to a subsequence or replaced by a polynomial: for integer-coefficient polynomials $Q$, Bourgain (1989) proved that $(1/N)\sum_{n\le N}f(T^{Q(n)}x)$ converges almost everywhere for $f\in L^r$, $r>1$, and for real-coefficient polynomials $P$ he proved the same for the averages along the integer parts $\lfloor P(n)\rfloor$ for $f\in L^\infty$. The standard route to such results is a maximal inequality, an a priori bound on $\sup_N|A_N f|$ in $L^r$, from which almost-everywhere convergence follows; the $L^1$ case, which would follow from a weak-type $(1,1)$ bound (an estimate $\mu(\sup_N|A_N f|>\lambda)\le C\|f\|_{L^1}/\lambda$), was unavailable for the thin exponent sets such as the squares and the primes, and the paper states that no $L^1$ theory could be developed.

Bourgain simultaneously isolated an abstract operator-theoretic question underlying all such results. A positive isometry of $L^2(\mu)$ is a linear operator $T$ with $\|Tf\|_{L^2}=\|f\|_{L^2}$ for all $f$ that maps nonnegative functions to nonnegative functions; composition $f\mapsto f\circ\phi$ with $\phi$ measure-preserving is the basic example, but the class is larger. For a sequence $(T_n)_{n\ge1}$ of commuting positive isometries of $L^2(\mu)$, write $A_N f=(1/N)\sum_{n\le N}T_n f$. The $\delta$-metrical entropy number $M_\delta(S)$ of a subset $S$ of $L^2(\mu)$ is the least number of closed $L^2$-balls of radius $\delta$ needed to cover $S$; it counts how many essentially different values a family takes. Bourgain proved in 1988 that almost-everywhere convergence of $A_N f$, even assumed only for bounded $f$, forces, for each $\delta>0$, a finite bound $M_\delta(\{A_N f\})<G(\delta)$, and that this bound must be uniform when $f$ ranges over the $L^2$ unit ball. The source paper records this in its final section as a necessary condition and observes that it is often effective in disproving convergence: it recovers Marstrand's counterexample to Khinchin's conjecture on strong uniform distribution (bounded measurable functions on the circle whose averages $(1/N)\sum_{n\le N}f(nx)$ diverge almost everywhere) and underpins recent divergence results for Khintchin-type averages. The converse — whether bounded orbit entropy guarantees convergence — has never been proved, and no other criterion, both necessary and sufficient, is known for commuting positive isometries of an infinite-dimensional $L^2(\mu)$.

Problem Statement

Let $(X,\mathcal B,\mu)$ be a probability space with $\dim L^2(\mu)=\infty$, and let $(T_n)_{n\ge1}$ be commuting positive isometries of $L^2(\mu)$: each $T_n$ is linear, $\|T_n f\|_{L^2}=\|f\|_{L^2}$ for all $f$, $T_n f\ge0$ whenever $f\ge0$, and $T_nT_m=T_mT_n$ for all $n,m$; the infinite-dimensionality excludes the degenerate case, in which the uniform entropy bound below holds automatically for every family. For $f\in L^2(\mu)$ define

$A_N f=\frac{1}{N}\sum_{n\le N}T_n f.$

Known necessary condition (Bourgain 1988, reproduced as Section 9(3) of the source): if $A_N f$ converges almost everywhere as $N\to\infty$ for every $f\in L^\infty(\mu)$, then for each $\delta>0$ there is $G(\delta)<\infty$ such that

$M_\delta(\{A_N f:N\ge1\})<G(\delta),$

where $M_\delta(S)$ is the least number of closed $L^2$-balls of radius $\delta$ covering the set $S$; the bound must be uniform in $f$ over the unit ball $\{\|f\|_{L^2}\le1\}$.

The problem is the converse and the resulting characterization, and the sub-questions are logically linked — because necessity is known, a positive answer to (i) yields the criterion (iii), while a negative answer to (i) is accompanied by the separation (ii):

(i) Sufficiency: does the uniform entropy bound above, assumed for the family $(T_n)$, imply that $A_N f(x)$ converges for almost every $x$ and every $f\in L^2(\mu)$?

(ii) Separation: if the answer to (i) is negative, exhibit a probability space with $\dim L^2(\mu)=\infty$, commuting positive isometries $(T_n)$ of its $L^2$ satisfying the uniform entropy bound, and a function $f\in L^2(\mu)$ such that $A_N f$ fails to converge on a set of positive measure.

(iii) Criterion: determine a condition on the family $(T_n)$ — entropy-type or otherwise — that is both necessary and sufficient for almost-everywhere convergence of $A_N f$ for every $f\in L^2(\mu)$. A proposed condition counts as a criterion only if it is a nontrivial structural condition on the family $(T_n)$ and the space, not a restatement of the convergence it is meant to characterise.

References

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2. Jean Bourgain, 'Almost sure convergence and bounded entropy', Israel Journal of Mathematics 63 (1988) 79-97, DOI 10.1007/BF02765022, https://doi.org/10.1007/BF02765022
3. Máté Wierdl, 'Pointwise ergodic theorem along the prime numbers', Israel Journal of Mathematics 64 (1988) 315-336, DOI 10.1007/BF02882425, https://doi.org/10.1007/BF02882425
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