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) optimal coefficients if, for any  nonzero $x\in \mathbb{F}_p$, one has \[ |x| |a_1 x| \dots |a_d x| \ge \frac{p^d}{M}.\]
    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 the above 
condition is found, then any function $f:[0,1]^{d+1}  \to \mathbb{R}$ 
can be integrated using the formula
    \[\left | \int_{[0,1]^{d+1}} f(x)\,dx - \frac{1}{p} \sum_{x=0}^{p-1} 
f\left(\frac{x}{p}, \left\lbrace \frac{a_1 x}{p} \right\rbrace, \dots, \left\lbrace 
\frac{a_d x}{p} \right\rbrace \right) \right| \ll \frac{M \log^d p \cdot 
\mathrm{V}(f)}{p}\]
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
\[\frac{a}{p} = \cfrac{1}{c_1 +\cfrac{1}{c_2 +\cdots +\cfrac{1}{c_s}}} \,,\qquad \text{all} \quad c_j \quad \text{are bounded.}\]

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.