Character Sums Over Bohr Sets

Hanson, Brandon

Canadian mathematical bulletin, Tome 58 (2015) no. 4, pp. 774-786

Résumé

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to the classical character sum bounds of Pólya–Vinogradov and Burgess. These estimates are applied to obtain results on recurrence $\bmod \,p$ by special elements.

Détail

DOI : 10.4153/CMB-2015-036-2

Mots-clés : 11L40, 11T24, 11T23, character sums, Bohr sets, finite fields

Bibliographie

[B] [B] Bourgain, J., On triples in arithmetic progression. Geom. Funct. Anal. 9(1999), 968–984. http://dx.doi.Org/10.1007/s000390050105 Google Scholar

[BGK] [BGK] Bourgain, J., Glibichuk, A. A., and Konyagin, S. V., Estimates for the number of sums and products and for exponential sums infields of prime order. J. London Math. Soc. (2) 73(2006), no. 2,380-398. http://dx.doi.Org/10.1112/S0024610706022721 Google Scholar

[BKT] [BKT] Bourgain, J., Katz, N., and Tao, T., A sum-product estimate in finite fields, and applications. Geom. Funct. Anal. 14(2004), no. 1, 27–57. http://dx.doi.Org/10.1007/s00039-004-0451-1 Google Scholar

[C] [C] Chang, M.-C., On a question of Davenport and Lewis and new character sum bounds infinite fields. Duke Math. J. 145(2008), no. 3, 409–442. http://dx.doi.Org/10.1215/00127094-2008-056 Google Scholar

[CLR] [CLR] Croot, E., Lyall, N., and Rice, A., A purely combinatorial approach to simultaneous polynomial recurrence modulo 1. arxiv:1307.0779 Google Scholar

[G] [G] Garaev, M. Z., An explicit sum-product estimate in W. Int. Math. Res. Not. IMRN 2007, no. 11, Art. ID rnmO35. Google Scholar

[GT] [GT] Green, B. and Tao, T., New bounds for Szemerédi's theorem. II. A new bound for r4(N). In: Analytic number theory, Cambridge University Press, Cambridge, 2009, pp. 180–204. Google Scholar

[IK] [IK] Iwaniec, H. and Kowalski, E., Analytic number theory. American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. Google Scholar

[KS] [KS] Katz, N. H. and C.-Y. Shen, A slight improvement to Garaev's sum product estimate. Proc. Amer. Math. Soc. 136(2008), no. 7, 2499–2504. http://dx.doi.Org/10.1090/S0002-9939-08-09385-4 Google Scholar

[K] [K] Konyagin, S. V., Estimates for character sums in finite fields. (Russian) Mat. Zametki 88(2010), no. 4,529–542; translation in Math. Notes 88(2010), no. 3-4, 503–515. http://dx.doi.Org/10.4213/mzm8852 Google Scholar

[LN] [LN] Lidl, R. and Neiderreiter, H., Finite fields. Encyclopedia of Mathematics and its Applications, 20, Cambridge University Press, Cambridge, 1997. Google Scholar

[LM] [LM] Lyall, N. and Magyar, A., Simultaneous polynomial recurrence. Bull. Lond. Math. Soc. 43(2011), no. 4, 765–785. http://dx.doi.Org/1 0.1112/blms/bdrO11 Google Scholar

[P] [P] Paley, R. E. A. C., A theorem on characters. J. Lond. Math. Soc. Sl-7(1932), no. 1, 28. http://dx.doi.Org/10.1112/jlms/s1-7.1.28 Google Scholar

[RNRS] [RNRS] Roche-Newton, O., Rudnev, M., and Shkredov, I., New sum-product type estimates over finite fields. arxiv:1408.0542v1 Google Scholar

[R] [R] Rudnev, M., An improved sum-product inequality infields of prime order. Int. Math. Res. Not. IMRN 2012, no. 16, 3693–3705. Google Scholar

[Sch] [Sch] Schmidt, W. M., Small fractional parts of polynomials. Regional Conference Series in Mathematics, 32, American Mathematical Society, 1977. Google Scholar

[Sh] [Sh] Shao, X., On character sums and exponential sums over generalized arithmetic progressions. Bull. Lond. Math. Soc. 45(2013), no. 3, 541–550. http://dx.doi.Org/10.1112/blms/bds11 5 Google Scholar

[TV] [TV] Tao, T. and Vu, V., Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006. http://dx.doi.Org/!0.1017/CBO9780511755149 Google Scholar