Short incomplete Gauss sums and rational points on metaplectic horocycles

Abstract

In this paper, we investigate the limiting behavior of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2, ℝ).

Translates of rational points along expanding closed horocycles on the modular surface

“…one can extend this equidistribution result to the range α ∈ (1, 2); this improves the previous work of Demirci Akarsu [5,Theorem 2] which confirms equdistribution of {R pr n (0, c/n α )} n∈N for α ∈ ( 3 2 , 2). Here j ∈ (Z/nZ) × denotes the multiplicative inverse of j ∈ (Z/nZ) × .…”

“…Indeed, it was shown by Luethi [ 24 ] that if for some and some , then becomes equidistributed on with respect to as . Moreover, under the simple symmetry relation that for and one can extend this equidistribution result to the range ; this improves the previous work of Demirci Akarsu [ 5 , Theorem 2] which confirms equdistribution of for . Here denotes the multiplicative inverse of .…”

“…The equidistribution for the case was later proved by Einsiedler–Luethi–Shah [ 8 ]; Jana [ 16 , Theorem 1] recently gave an alternative spectral proof to this equidistribution result. We also mention that both [ 5 , Theorem 2] and [ 16 , Theorem 1] are valid in the same setting as [ 8 ], namely, on the product of the unit tangent bundle of the modular surface and a torus. When the equidistribution fails as the aforementioned symmetry implies that is always trapped in the closed horocycle .…”

Limit laws for rational continued fractions and value distribution of quantum modular forms

“…Another tentative approach to Theorem 2.1 consists in obtaining a limiting distribution result for $$\sum _{n\geqslant 1}\tau (n){\rm e}(nz)$$, where $$z=x+iy$$; $$x\in [0,1]$$ is chosen at random and $$y\rightarrow 0^+$$, and transferring these properties to its discrete counterpart $$D(\frac{1}{2}, x)\approx \sum _{n\leqslant q^2} \tau (n){\rm e}(nx)n^{-1/2}$$, for $$x=a/q\in {\mathbb {Q}}$$, by Fourier expansion. This method is employed for the incomplete Gauss sum in [36], based on [81]. In our setting, however, connecting the continuous and the discrete averages raises several additional difficulties, among which the problem of dealing with a divergent second moment as well as those coming from considering a central value rather than an object corresponding to an $$L$$‐value off the line.…”

Random process generated by the incomplete Gauss sums

“…This study extends the author and Marklof's [2] work on the value distribution of long incomplete Gauss sums. The above-mentioned work is later extended to the short interval case of incomplete Gauss sums by the author [3]. The classical examples of incomplete Gauss sums were also studied in the literature for many others [5,9,12,13,14].…”