Smooth discrepancy and Littlewood's conjecture
Smooth discrepancy and Littlewood's conjecture
Given $\boldsymbol{\alpha} \in [0,1]^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n \boldsymbol{\alpha} \,\mathrm{mod}\, 1)_{n\geq 1}$. We find that it can be smaller than the classical discrepancy of $\textbf{any}$ sequence when $d \le 2$, and can even be bounded in the case $d=1$. To achieve this, we establish …