  An important result of H. Weyl states that for every sequence
$\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of
fractional parts of $\left(a_{n} \alpha \right)_{n\geq 1}$ is uniformly
distributed modulo one for almost all $\alpha$. However, in general it is a
very hard problem to calculate the precise order of convergence of the
discrepancy of $\left(\left\{a_{n} \alpha\right\}\right)_{n \geq 1}$ for almost
all $\alpha$. In particular it is very difficult to give sharp lower bounds for
the speed of convergence. Until now this was only carried out for lacunary
sequences $\left(a_{n}\right)_{n \geq 1}$ and for some special cases such as
the Kronecker sequence $\left(\left\{n \alpha\right\}\right)_{n \geq 1}$ or the
sequence $\left(\left\{n^2 \alpha\right\}\right)_{n \geq1}$. In the present
paper we answer the question for a large class of sequences
$\left(a_{n}\right)_{n \geq 1}$ including as a special case all polynomials
$a_{n} = P\left(n\right)$ with $P \in \mathbb{Z} \left[x\right]$ of degree at
least 2.
