Author

N. Shulga

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Preprint Aug 2026

Nine-distance theorem and growth of best-approximation denominators

We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we p...

N. Shulga · 4 citations · ⚡3

Preprint Aug 2026

The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension

The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|n\xi\|\,\|n\zeta\|=0 $$ for all real $\xi,\zeta$. It is proven to hold for almost every pair $(\xi,\zeta)$. Schleischitz, however, has recently disproved the...

N. Shulga · 1 citation