Abstract
We consider quadratic Weyl sums SN(x;c,α)=∑n=1Nexp{2πi((12n2+cn)x+αn)} for c= α= 0 (the rational case) or (c, α) ∉ Q2 (the irrational case), where x∈ R is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. The limiting distribution in the complex plane of 1NSN(x;c,α) as N→ ∞ was described in Marklof (Duke Math J 97(1):127–153, 1999) [respectively Cellarosi and Marklof in Proc Lond Math Soc 113(6):775–828, 2016)] in the rational (resp. irrational) case. According to the limiting distribution, the probability of landing outside a ball of radius R is known to be asymptotic to 4log2π2R-4(1+o(1)) in the rational case and to 6π2R-6(1+O(R-12/31)) in the irrational case, as R→ ∞. In this work we refine the technique of Cellarosi and Marklof (2016) to improve the known tail estimates to 4log2π2R-4(1+Oε(R-2+ε)) and 6π2R-6(1+Oε(R-2+ε)) for every ε> 0. In the rational case, we rely on the equidistribution of a rational horocycle lift to a torus bundle over the unit tangent bundle to the classical modular surface. All the constants implied by the Oε-notations are made explicit. © 2023, The Author(s), under exclusive licence to Unione Matematica Italiana.