Intrinsic diophantine approximation on $S^3$ and sums of Kloosterman sums

Abstract: Let $S^3$ denote the unit sphere in $\mathbb{R}^4$. In a letter
about the efficiency of a universal set of quantum gates, Sarnak raised
the question of how well one can approximate points on $S^3$ by rational
points of small height. In particular, given $r \in \mathbb{N}$, how large
does $\epsilon$ need to be so that any point on $S^3$ can be approximated
within $\epsilon$ to a point of the form $\mathbf{x}/r$, with $\mathbf{x}
\in \mathbb{Z}^4$? Using the smooth $\delta$-function form of the
Hardy-Littlewood circle method, Nasser Sardari showed that $\epsilon \gg
r^{-1/3+o(1)}$ is sufficient. In this talk, I will describe how a variant
of the Linnik conjecture, which concerns sums of Kloosterman sums, allows
us to take a smaller value of $\epsilon$. Joint work with Tim Browning and
Raphael Steiner.