Improved tail estimates for the distribution of quadratic Weyl sums. (English) Zbl 1523.37009
Techniques for tail estimates (that is, the asymptotic probability of falling outside a ball of radius \(R\)) for normalized Weyl sums of the form
\[
(1/\sqrt{N})\sum_{n=1}^{N}\exp \bigl(2\pi{\mathrm{i}}((n^2/2+cn)x+\alpha n)\bigr)
\]
obtained by F. Cellarosi and J. Marklof [Proc. Lond. Math. Soc. (3) 113, No. 6, 775–828 (2016; Zbl 1423.11145)] are refined to improve the estimates.
Reviewer: Thomas B. Ward (Durham)
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 37A44 | Relations between ergodic theory and number theory |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
Citations:
Zbl 1423.11145References:
| [1] | Athreya, J.S., Chaika, J., Lelièvre, S.: The gap distribution of slopes on the golden L. In: Bhattacharya, S., Das, T., Ghosh, A., Shah, R. (eds.) Recent trends in ergodic theory and dynamical systems, volume 631 of Contemp. Math., pp. 47-62. American Mathematical Society, Providence (2015) · Zbl 1332.37027 |
| [2] | Boca, FP; Cobeli, C.; Zaharescu, A., Distribution of lattice points visible from the origin, Commun. Math. Phys., 213, 2, 433-470 (2000) · Zbl 0989.11049 · doi:10.1007/s002200000250 |
| [3] | Boca, FP; Heersink, B.; Spiegelhalter, P., Gap distribution of Farey fractions under some divisibility constraints, Integers, 13, A44, 15 (2013) · Zbl 1290.11035 |
| [4] | Boca, FP; Zaharescu, A., The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit, Commun. Math. Phys., 269, 2, 425-471 (2007) · Zbl 1143.37002 · doi:10.1007/s00220-006-0137-7 |
| [5] | Cellarosi, F., Limiting curlicue measures for theta sums, Ann. Inst. Henri Poincaré Probab. Stat., 47, 2, 466-497 (2011) · Zbl 1233.37026 · doi:10.1214/10-AIHP361 |
| [6] | Cellarosi, F.; Marklof, J., Quadratic weyl sums, automorphic functions and invariance principles, Proc. Lond. Math. Soc., 113, 6, 775-828 (2016) · Zbl 1423.11145 · doi:10.1112/plms/pdw038 |
| [7] | Cellarosi, F., Osman, T.: Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters. arXiv:2210.09838 |
| [8] | Demirci Akarsu, E., Short incomplete Gauss sums and rational points on metaplectic horocycles, Int. J. Number Theory, 10, 6, 1553-1576 (2014) · Zbl 1316.11072 · doi:10.1142/S1793042114500444 |
| [9] | Demirci Akarsu, E.; Marklof, J., The value distribution of incomplete Gauss sums, Mathematika, 59, 2, 381-398 (2013) · Zbl 1316.11073 · doi:10.1112/S0025579312001179 |
| [10] | Dolgopyat, D.; Fayad, B., Deviations of ergodic sums for toral translations I. Convex bodies, Geom. Funct. Anal., 24, 1, 85-115 (2014) · Zbl 1308.37007 · doi:10.1007/s00039-014-0254-y |
| [11] | Dolgopyat, D.; Fayad, B., Deviations of ergodic sums for toral translations II. Boxes, Publ. Math. Inst. Hautes Études Sci., 132, 293-352 (2020) · Zbl 1473.37012 · doi:10.1007/s10240-020-00120-2 |
| [12] | Dudley, RM, Real Analysis and Probability. Cambridge Studies in Advanced Mathematics (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1023.60001 · doi:10.1017/CBO9780511755347 |
| [13] | Elkies, ND; McMullen, CT, Gaps in \({\sqrt{n}}\;{\rm mod} \;1\) and ergodic theory, Duke Math. J., 123, 1, 95-139 (2004) · Zbl 1063.11020 · doi:10.1215/S0012-7094-04-12314-0 |
| [14] | Gut, A., Probability: A Graduate Course. Springer Texts in Statistics (2013), New York: Springer, New York · Zbl 1267.60001 · doi:10.1007/978-1-4614-4708-5 |
| [15] | Kowalski, E.; Sawin, WF, Kloosterman paths and the shape of exponential sums, Compos. Math., 152, 7, 1489-1516 (2016) · Zbl 1419.11134 · doi:10.1112/S0010437X16007351 |
| [16] | Lion, G.; Vergne, M., The Weil Representation. Maslov Index and Theta Series, Volume 6 of Progress in Mathematics (1980), Boston: Birkhäuser, Boston · Zbl 0444.22005 · doi:10.1007/978-1-4684-9154-8 |
| [17] | Marklof, J., Limit theorems for theta sums, Duke Math. J., 97, 1, 127-153 (1999) · Zbl 0965.11036 · doi:10.1215/S0012-7094-99-09706-5 |
| [18] | Marklof, J., Almost modular functions and the distribution of \(n^2x\) modulo one, Int. Math. Res. Not., 39, 2131-2151 (2003) · Zbl 1056.11022 · doi:10.1155/S1073792803130292 |
| [19] | Marklof, J., Spectral theta series of operators with periodic bicharacteristic flow, Ann. Inst. Fourier (Grenoble), 57, 7, 2401-2427 (2007) · Zbl 1133.35075 · doi:10.5802/aif.2338 |
| [20] | Marklof, J., The asymptotic distribution of Frobenius numbers, Invent. Math., 181, 1, 179-207 (2010) · Zbl 1200.11022 · doi:10.1007/s00222-010-0245-z |
| [21] | Marklof, J.; Strömbergsson, A., The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Ann. Math. (2), 172, 3, 1949-2033 (2010) · Zbl 1211.82011 · doi:10.4007/annals.2010.172.1949 |
| [22] | Marklof, J.; Strömbergsson, A., The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates, Geom. Funct. Anal., 21, 3, 560-647 (2011) · Zbl 1242.82036 · doi:10.1007/s00039-011-0116-9 |
| [23] | Marklof, J.; Strömbergsson, A., Power-law distributions for the free path length in Lorentz gases, J. Stat. Phys., 155, 6, 1072-1086 (2014) · Zbl 1304.82066 · doi:10.1007/s10955-014-0935-9 |
| [24] | Nándori, P.; Szász, D.; Varjú, T., Tail asymptotics of free path lengths for the periodic Lorentz process: on Dettmann’s geometric conjectures, Commun. Math. Phys., 331, 1, 111-137 (2014) · Zbl 1379.37079 · doi:10.1007/s00220-014-2086-x |
| [25] | Sarnak, P., Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Commun. Pure Appl. Math., 34, 6, 719-739 (1981) · Zbl 0501.58027 · doi:10.1002/cpa.3160340602 |
| [26] | Uyanik, C.; Work, G., The distribution of gaps for saddle connections on the octagon, Int. Math. Res. Not. IMRN, 18, 5569-5602 (2016) · Zbl 1404.37039 · doi:10.1093/imrn/rnv317 |
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