Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$
Leonidas Daskalakis
TL;DR
This work proves the first pointwise convergence result for bilinear ergodic averages taken along deterministic sparse orbits with modulation invariance, covering averages $ rac{1}{N} extstyle\sum_{n=1}^N f(T^{ loor{n^c}}x) g(T^{- loor{n^c}}x)$ for $c o(1,23/22)$ and more generally orbits $ loor{h(n)}$ with $h ext{ in }\\mathcal{R}_c$. The authors reduce the problem to a main term that matches Bourgain's bilinear averages plus a controlled error term, then develop a Fourier-analytic and Gowers-norm framework to bound the error. A key technical achievement is establishing $L^1$-type decay for a dyadic kernel via a $U^3$-norm analysis and exponential-sum estimates, enabling a Calderón-type transference to the ergodic setting. The results extend the scope of pointwise bilinear convergence to sparse deterministic orbits and lay groundwork for further exploration of modulation-invariant phenomena in multilinear ergodic theory.
Abstract
For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},μ,T)$ we prove that for any $f,g\in L^{\infty}_μ(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor n^c\rfloor}x)\quad\text{converge for $μ$-a.e. $x\in X$.} \] In fact, we consider more general sparse orbits $(\lfloor h(n)\rfloor,-\lfloor h(n)\rfloor)_{n\in\mathbb{N}}$, where $h$ belongs to the class of the so-called $c$-regularly varying functions. This is the first pointwise result for bilinear ergodic averages taken along deterministic sparse orbits where modulation invariance is present.