Almost Sure Bounds for Discrepancies of Linear Forms on the Circle

As a generalization of irrational rotations and a dual case of higher-dimensional Kronecker sequences, we study the discrepancy of sequences of linear forms on the circle. Given irrationals $\alpha_1,\dots,\alpha_d$, consider the set of $N^d$ points $\{k_1\alpha_1+\cdots+k_d\alpha_d \mod 1 : 1\le k_j\le N\}$. We prove that for a full-measure set of vectors $(\alpha_1,\dots,\alpha_d)\in\mathbb{R}^d$, the maximal discrepancy of these points relative to intervals in $[0,1)$ has the optimal principal order $(\log N)^d$, up to powers of $\log\log N$. This result provides a nearly sharp dual analogue, in the setting of linear forms, to Beck's celebrated theorem on multidimensional Kronecker sequences (Ann. of Math., 1994). The proof combines Fourier analysis, metric multiplicative Diophantine estimates, and a duality argument which reduces certain lattice-counting errors to Beck's discrepancy theorem.