Joint IAS/PU Groups and Dynamics Seminar

Let $p$ be a prime number, $d$ be a positive integer, and $M\ge 1$ be a real parameter. 
A tuple $(a_1,\dots, a_d) \in \mathbb{F}^d_p$ is called a tuple of (Korobov) $\it optimal$ $\it coefficients$ if, for any  nonzero $x\in \mathbb{F}_p$ one has 
\begin{equation}\label{f:*}
    |x| |a_1 x| \dots |a_d x| \ge \frac{p^d}{M} 
    \,.
\end{equation}
Here for $y\in\mathbb F_p$ we write $|y|$ for the absolute value of the representative of $y$ in $(-p/2,p/2]$. These coefficients arise naturally in problems of numerical integration. Namely, if a tuple $(a_1, \dots, a_d)$ satisfying condition (the above condition) is found, then any function $f:[0,1]^{d+1}  \to \mathbb{R}$ can be integrated using the formula
    \[
\left| \int_{[0,1]^{d+1}} f(x)\,dx - \frac{1}{p} \sum_{x=0}^{p-1} f\left(\frac{x}{p}, \left\{ \frac{a_1 x}{p} \right\}, \dots, \left\{ \frac{a_d x}{p} \right\} \right) \right| \ll \frac{M \log^d p \cdot \mathrm{V}(f)}{p} \,,
\]
where $\mathrm{V}(f)$ is the  Hardy--Krause variation of the function $f$. Korobov (1959--1963) proved that the case $M=O(\log^d p)$ is always realizable, whereas the special case $d=1$, $M=O(1)$ is equivalent to the well-known Zaremba conjecture (1972): for any $p$ one can find $1\le a<p$ such that   

\[  \frac{a}{p} =    \cfrac{1}{c_1 +\cfrac{1}{c_2 +\cdots +\cfrac{1}{c_s}}} , \text{  all } c_j \text{ are bounded.}\] 

For $d>1$ and $M =o (\log^d p)$, only a few results are known. 

We give an overview of the problems in this area and describe recent advances and connections to other topics in number theory and combinatorics. We also discuss the connection between this topic and McMullen's arithmetic chaos conjecture.