# Mathematics Colloquia and Seminars

Let ${\bf X_1}=(X,\Sigma_1,m_1, \tau)$ be a dynamical system. Let also $1\le p,q\le \infty$ be such that $\frac1p+\frac1q\le 1$. The Return Times Theorem proved by Bourgain asserts the following: For each function $f\in L^{p}(X)$ there is a universal set $X_0\subseteq X$ with $m_1(X_0)=1$, such that for each second dynamical system ${\bf Y}=(Y,\Sigma_2,m_2,\sigma)$, each $g\in L^{q}(Y)$ and each $x\in X_0$, the averages $$\frac1{N}\sum_{n=0}^{N-1}f(\tau^nx)g(\sigma^ny)$$ converge $m_2$ almost everywhere.
We show how to break the duality in this theorem. More precisely, we prove that the result remains true if $p>1$ and $q\ge 2$. We emphasize the strong connections between this result and the Carleson-Hunt theorem on the convergence of the Fourier series. We also prove similar results for the analog of Bourgain's theorem for signed averages, where no positive results were previously known. This is joint work with Michael Lacey, Terence Tao and Christoph Thiele.