Metric number theory, lacunary series and systems of dilated functions

By a classical result of Weyl, for any increasing sequence $(n_k{)}_{k\ge 1}$ of integers the sequence of fractional parts $(\{n_{k}x\}{)}_{k\ge 1}$ is uniformly distributed modulo 1 for almost all $x\in [0,1]$. Except for a few special cases, e.g. when $n_k=k,k\ge 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $({n_k x}){k \geq 1}$isonlyknowninafewspecialcases,forexamplewhen$(n_k){k \geq 1}$isa(Hadamard)lacunarysequence,thatiswhen$n_{k+1}/n_k \geq q > 1, k \geq 1$\mathrm{.}Inthiscaseofquicklyincreasing$(n_k){k \geq 1}$thesystem$({n_k x}){k \geq 1}$(or,moregeneral,$(f(n_k x))_{k \geq 1}$fora1-periodicfunction$f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena. Without any growth conditions on $(n_k{)}_{k\ge 1}$ the situation becomes much more complicated, and the system $(f(n_{k}x){)}_{k\ge 1}$ will typically fail to satisfy probabilistic limit theorems. An important problem which remains is to study the almost everywhere convergence of series ${\sum }_{k=1}^{\mathrm{\infty }}{c}_{k}f(kx)$, which is closely related to finding upper bounds for maximal $L^2$-norms of the form $${\int }_0^1(\underset{1\le M\le N}{\max }\mathrm{\mid }\sum _{k=1}^M{c}_{k}f(kx){\mathrm{\mid }}^{2}dx\mathrm{.}$$ The most striking example of this connection is the equivalence of the Carleson convergence theorem and the Carleson–Hunt inequality for maximal partial sums of Fourier series. For general functions $f$ this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors.