# Difference between revisions of "Littlewood problem"

The Littlewood problem for compatible Diophantine approximations is the question of the existence, for any real numbers $\alpha,\,\beta,\,\epsilon>0$, of a natural number $n$ such that $n \cdot \Vert n\alpha \Vert \cdot \Vert n\beta \Vert < \epsilon$, where $\Vert x \Vert$ is the distance from $x$ to the nearest integer. In certain cases, for example for rational $\alpha$ and $\beta$, and for numbers $\alpha$ and $\beta$ one of which can be represented by a continued fraction with non-negative elements, the Littlewood problem has an affirmative answer.
The Littlewood problem for integrals is the problem whether for an arbitrary increasing sequence $M$ of natural numbers $(m_k)$, $k=1,2,\ldots$, one has \begin{equation}\label{eq:1} \int_0^1 \left\vert { \sum_{k=1}^n \exp(2\pi i m_k x) }\right\vert dx > f(n) \end{equation} with $f(n) = C \log n$, where $C > 0$ is an absolute constant and $n > n_0$. The following estimates have been obtained: either weaker estimates in comparison with \eqref{eq:1} for arbitrary sequences $M$, or estimates close to \eqref{eq:1} or even coinciding with this estimate, but for special sequences $M$.