The Lov\'asz conjecture holds for moderately dense Cayley graphs

We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lov\'asz conjecture and improves upon the previous best result of this form due to Christofides, Hladk\'y, and M\'ath\'e from 2014 concerning graphs with $d\geq \varepsilon n$. Our proof avoids the use of Szemer\'edi's regularity lemma and relies instead on an efficient arithmetic regularity lemma specialised to Cayley graphs.