Title
Asymptotic gaps in the Farey sequence of an imaginary quadratic number field
Abstract
We will quickly present a well known result on the gaps of real Farey fractions: with a standard rescaling, they converge in average towards the density of R. R. Hall (1970). He obtained this result with number theoretic arguments, then it was found again by others using arguments from dynamics, in particular by J. Marklof (2013) using homogeneous dynamics on SL_2(R) / SL_2(Z).
Let K be a quadratic imaginary field and denote by O_K its ring of integers. After adapting the method of Marklof and lifting a joint equidistribution result of J. Parkkonen and F. Paulin (2022), we will obtain a result on the asymptotic distribution of gaps in the fractions of elements of O_K. The homogeneous dynamics will be set on the double quotient M \ PSL_2(C) / PSL_2(O_K) (with M the maximal diagonal compact subgroup). We will prove the existence of an asymptotic density for these gaps, the cumulative distribution function will be described geometrically, and from this description we will be able to estimate its tail distribution in the two particular cases of Gaussian and Eisenstein fractions.
