arXiv subjects
Publications and source records attributed to Zouhair Ouaggag.
We study intrinsic Diophantine approximation on the sphere $S^d$, measured by the quality of the best rational approximation available at a given height. We show that, for $d\geq 3$ and along sufficiently sparse sequences of heights, these quantities have a Weibull limit distribution and the associated counting functions are asymptotically Poissonian.
We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an error term of square root order, analogous to the optimal estimate in the Euclidean setting. We also show that the counting function has vanishing third and higher correlations and derive a Central Limit Theorem describing its fluctuations. We prove these results using arguments from homogeneous dynamics on the space of orthogonal lattices, in particular effective multiple equidistribution of all orders, which we establish for spherical averages and which could be useful for other applications.
We prove an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$. We use homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence estimates for the Siegel transform, developping on recent results of Alam-Ghosh and Kleinbock-Merrill.