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Ben Krause, Mariusz Mirek, and I have uploaded to the arXiv our paper Pointwise ergodic theorems for non-conventional bilinear polynomial averages. This paper is a contribution to the decades-long program of extending the classical ergodic theorems to “non-conventional” ergodic averages. Here, the focus is on pointwise convergence theorems, and in particular looking for extensions of the pointwise ergodic theorem of Birkhoff:
Theorem 1 (Birkhoff ergodic theorem) Letbe a measure-preserving system (by which we mean
is a
-finite measure space, and
is invertible and measure-preserving), and let
for any
. Then the averages
converge pointwise for
-almost every
.
Pointwise ergodic theorems have an inherently harmonic analysis content to them, as they are closely tied to maximal inequalities. For instance, the Birkhoff ergodic theorem is closely tied to the Hardy-Littlewood maximal inequality.
The above theorem was generalized by Bourgain (conceding the endpoint , where pointwise almost everywhere convergence is now known to fail) to polynomial averages:
Theorem 2 (Pointwise ergodic theorem for polynomial averages) Letbe a measure-preserving system, and let
for any
. Let
be a polynomial with integer coefficients. Then the averages
converge pointwise for
-almost every
.
For bilinear averages, we have a separate 1990 result of Bourgain (for functions), extended to other
spaces by Lacey, and with an alternate proof given, by Demeter:
Theorem 3 (Pointwise ergodic theorem for two linear polynomials) Letbe a measure-preserving system with finite measure, and let
,
for some
with
. Then for any integers
, the averages
converge pointwise almost everywhere.
It has been an open question for some time (see e.g., Problem 11 of this survey of Frantzikinakis) to extend this result to other bilinear ergodic averages. In our paper we are able to achieve this in the partially linear case:
Theorem 4 (Pointwise ergodic theorem for one linear and one nonlinear polynomial) Letbe a measure-preserving system, and let
,
for some
with
. Then for any polynomial
of degree
, the averages
converge pointwise almost everywhere.
We actually prove a bit more than this, namely a maximal function estimate and a variational estimate, together with some additional estimates that “break duality” by applying in certain ranges with , but we will not discuss these extensions here. A good model case to keep in mind is when
and
(which is the case we started with). We note that norm convergence for these averages was established much earlier by Furstenberg and Weiss (in the
case at least), and in fact norm convergence for arbitrary polynomial averages is now known thanks to the work of Host-Kra, Leibman, and Walsh.
Our proof of Theorem 4 is much closer in spirit to Theorem 2 than to Theorem 3. The property of the averages shared in common by Theorems 2, 4 is that they have “true complexity zero”, in the sense that they can only be only be large if the functions involved are “major arc” or “profinite”, in that they behave periodically over very long intervals (or like a linear combination of such periodic functions). In contrast, the average in Theorem 3 has “true complexity one”, in the sense that they can also be large if
are “almost periodic” (a linear combination of eigenfunctions, or plane waves), and as such all proofs of the latter theorem have relied (either explicitly or implicitly) on some form of time-frequency analysis. In principle, the true complexity zero property reduces one to study the behaviour of averages on major arcs. However, until recently the available estimates to quantify this true complexity zero property were not strong enough to achieve a good reduction of this form, and even once one was in the major arc setting the bilinear averages in Theorem 4 were still quite complicated, exhibiting a mixture of both continuous and arithmetic aspects, both of which being genuinely bilinear in nature.
After applying standard reductions such as the Calderón transference principle, the key task is to establish a suitably “scale-invariant” maximal (or variational) inequality on the integer shift system (in which with counting measure, and
). A model problem is to establish the maximal inequality
The first step is to understand when the single-scale estimate (2) can come close to equality. A key example to keep in mind is when ,
,
where
is a small modulus,
are such that
,
is a smooth cutoff to an interval
of length
, and
is also supported on
and behaves like a constant on intervals of length
. Then one can check that (barring some unusual cancellation) (2) is basically sharp for this example. A remarkable result of Peluse and Prendiville (generalised to arbitrary nonlinear polynomials
by Peluse) asserts, roughly speaking, that this example basically the only way in which (2) can be saturated, at least when
are supported on a common interval
of length
and are normalised in
rather than
. (Strictly speaking, the above paper of Peluse and Prendiville only says something like this regarding the
factors; the corresponding statement for
was established in a subsequent paper of Peluse and Prendiville.) The argument requires tools from additive combinatorics such as the Gowers uniformity norms, and hinges in particular on the “degree lowering argument” of Peluse and Prendiville, which I discussed in this previous blog post. Crucially for our application, the estimates are very quantitative, with all bounds being polynomial in the ratio between the left and right hand sides of (2) (or more precisely, the
-normalized version of (2)).
For our applications we had to extend the inverse theory of Peluse and Prendiville to an
theory. This turned out to require a certain amount of “sleight of hand”. Firstly, one can dualise the theorem of Peluse and Prendiville to show that the “dual function”
Using these inverse theorems (and the Ionescu-Wainger multiplier theory) one still has to understand the “major arc” portion of (1); a model case arises when are supported near rational numbers
with
for some moderately large
. The inverse theory gives good control (with an exponential decay in
) on individual scales
, and one can leverage this with a Rademacher-Menshov type argument (see e.g., this blog post) and some closer analysis of the bilinear Fourier symbol of
to eventually handle all “small” scales, with
ranging up to say
where
for some small constant
and large constant
. For the “large” scales, it becomes feasible to place all the major arcs simultaneously under a single common denominator
, and then a quantitative version of the Shannon sampling theorem allows one to transfer the problem from the integers
to the locally compact abelian group
. Actually it was conceptually clearer for us to work instead with the adelic integers
, which is the inverse limit of the
. Once one transfers to the adelic integers, the bilinear operators involved split up as tensor products of the “continuous” bilinear operator
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