10 problems

Let $(a_k)_{k\geq 1}$ be a lacunary sequence satisfying … where $\eta$ is transcendental. Let $S_n$ be the associated lacunary trigonometric sum, and let the corresponding independ…

Let $\eta>1$ be algebraic but not a Perron number, and consider a lacunary sequence of the form … Let $S_n$ denote the associated lacunary trigonometric sum, and let…

For $0<\beta\leq \tfrac12$, define $n_k=\lfloor e^{k^\beta}\rfloor$ and consider the random variable sequence $(\cos(2\pi n_kx))_{k\geq 1}$ with $x$ distributed on $(0,1)$. Erdős's…

Let $(n_k)_{k\geq 1}$ be a Hadamard lacunary sequence, meaning that there is a constant $q>1$ such that $n_{k+1}/n_k\geq q$ for all $k$. For $N\geq 1$, let $D_N(\{n_kx\})$ denote t…

Let $q\in\{2,3,\ldots\}$ and let $a_k=q^k$ for $k\in\mathbb{N}$. Let $I_q$ be the rate function in the large deviation principle for $(S_n/n)_{n\in\mathbb{N}}$, and let…

Let $(n_k)$ be a lacunary sequence of natural numbers, and let $f$ belong to the Orlicz class $L\log^+\log^+ L[0,2\pi)$. Almost-everywhere convergence along $(n_k)$ means that the…

Lacunary maximal-operator conjecture. The operators $A^{(Q,psi)}_{L,*}$ are bounded on $ell^p(Z^n)$ for

Let $(n_k)$ be a sequence of positive integers satisfying Hadamard's gap condition … For a real number $x$, let $D_N\{n_kx\}$ denote the discrepancy of the sequence $(n_kx)$ modulo…

Let $(n_k)_{k\geq 1}$ be a lacunary sequence, and consider the almost independent system $(\{n_kx\})_{k\geq 1}$ for almost every $x$. The Chung–Smirnov law of the iterated logarith…

Let $\{n_j:j\ge 1\}$ be a lacunary sequence of integers, meaning that … For a function $f$ on $\mathbb T$, let $S_{n_j}f$ denote its Walsh-Fourier partial sums, and write…