On Korobov's optimal coefficients

Abstract: Let $p$ be a prime number, $d$ be a positive integer, and $M\ge 1$ be a real parameter. A tuple $(a_1,\dots, a_d) \in \mathbf{F}^d_p$ is called a tuple of (Korobov) {\it optimal coefficients} if, for any nonzero $x\in \mathbf{F}_p$, the inequality$$ x|a_1 x| \dots |a_d x| \ge \frac{p^d}{M} $$ holds. These famous coefficients arise naturally in numerical integration problems. Namely, if a tuple $(a_1, \dots, a_d)$ satisfying the inequality is found, then any function $f:[0,1]^d \to \mathbf{R}$ can be integrated using the formula $$ \left| \int_{[0,1]^d} f(x)\,dx - \frac{1}{p} \sum_{x=1}^{p} f\left(\frac{a_1 x}{p}, \dots, \frac{a_d x}{p} \right) \right| \ll \frac{M\cdot \mathrm{V}(f)}{p} \,, $$ where $\mathrm{V}(f)$ is the Hardy--Krause variation of the function $f$. Korobov proved that the case $M=O((\log p)^{d-1})$ is always realizable, whereas the special case $d=1$, $M=O(1)$ is equivalent to the well-known Zaremba conjecture. For $d>1$ and arbitrary $M$, only a few results are known. In our talk, we will provide an overview of the problems in this area and describe recent advances and connections to other topics in number theory.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)