Title: Discrete (Random) Carleson Theorems with Monomial Modulations
Abstract: Let $\{X_m\}$ be an independent sequence of $\{0,1\}$ random variables (on a probability space $\Omega$) with expectations
\[ \Bbb E X_m = m^{\frac{1}{d}-1}, \ d \geq 3. \]
Define $a_m(\omega)$ to be the random counting function associated to the $\{ X_m \}$: $a_m$ is the smallest integer such that
\[ X_1 + \dots + X_{a_m} = m.\]
Note that by the strong law of large numbers, almost surely $\frac{a_m}{m^d} \to c_d \in (0,\infty)$, so that the $\{ a_m \}$ form a random model for the sequence of monomial powers.
We give sufficient conditions on $\Lambda \subset [0,1]$, met by certain types of Cantor sets, under which the random discrete maximal functions below are almost surely bounded on $\ell^2(\Bbb Z)$.
\[
\mathcal{C}_{d,\Lambda}^\omega f(x) :=
\begin{cases} \sup_{\lambda \in \Lambda} \left| \sum_{m \neq 0} \frac{e^{2\pi i \lambda a_{|m|}(\omega)}}{m} f (x-m) \right| &\mbox{if } d\geq 3 \ \text{ is even} \\
\sup_{\lambda \in \Lambda} \left| \sum_{m \neq 0} \frac{e^{2 \pi i \lambda sgn(m) a_{|m|}(\omega)}}{m} f (x-m) \right| &\mbox{if } d\geq 3 \ \text{ is odd.}
\end{cases} .\]
Our method of proof allows us to handle the deterministic maximal functions
\[
\mathcal{C}_{d,\Lambda} f(x) :=
\sup_{\lambda \in \Lambda} \left| \sum_{m \neq 0} \frac{e^{2\pi i \lambda m^d}}{m} f (x-m) \right|, \ d \geq 3 \]
as well (the $d=2$ case has been handled previously).
Joint work with Michael Lacey.