Smooth discrepancy and Littlewood's conjecture

Given $\mathit{\alpha }\in [0,1{]}^d$, we estimate the smooth discrepancy of the Kronecker sequence $(n\mathit{\alpha }\mathrm{m}\mathrm{o}\mathrm{d}1{)}_{n\ge 1}$. We find that it can be smaller than the classical discrepancy of $\mathbf{\text{any}}$ 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.