Colloquium abstracts

Vinay Kumaraswamy
TIFR, Mumbai
February 11, 2021

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