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 a novel deterministic analogue of Beck's local-to-global principle (Ann. of Math. 1994), which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation. This opens up a new avenue of attack for Littlewood's conjecture.
Categories: math.NT math.CA math.DS