On the distribution of sparse sequences in prime fields and applications

 Access to full text

1. W. D. Banks, A. Conflitti, J. B. Friedlander and I. E. Shparlinski, Exponential sums over Mersenne numbers. Compos. Math. 140 (2004), no. 1, 15–30. Zbl1060.11045MR2004121
2. W. D. Banks, M. Z. Garaev, F. Luca and I. E. Shparlinski, Uniform distribution of fractional parts related to pseudoprimes. Canad. J. Math. 61 (2009), no. 3, 481–502. Zbl1255.11040MR2514480
3. J. Bourgain, Estimates on exponential sums related to the Diffie–Hellman distributions. Geom. Funct. Anal. 15 (2005), no. 1, 1–34. Zbl1102.11041MR2140627
4. E. Croot, Sums of the form $1/x_1^k+\cdots +1/x_n^k$ modulo a prime. Integers 4 (2004), A20, 6 pp. Zbl1083.11019MR2116005
5. C. Elsholtz, The distribution of sequences in residue classes. Proc. Amer. Math. Soc. 130 (2002), no. 8, 2247–2250. Zbl1004.11052MR1896404
6. P. Erdős and M. R.  Murty, On the order of $a(modp)$. Proc. 5th Canadian Number Theory Association Conf., Amer. Math. Soc., Providence, RI, 1999, 87–97. Zbl0931.11034MR1684594
7. M. Z. Garaev, Upper bounds for the number of solutions of a diophantine equation. Trans. Amer. Math. Soc. 357 (2005), no. 6, 2527–2534. Zbl1114.11031MR2140449
8. M. Z. Garaev, The large sieve inequality for the exponential sequence ${\lambda }^{\left[O\left(n^{15/14+o\left(1\right)}\right)\right]}$ modulo primes. Canad. J. Math. 61 (2009), no. 2, 336–350. Zbl1179.11024MR2504019
9. M. Z. Garaev and Ka–Lam Kueh, Distribution of special sequences modulo a large prime. Int. J. Math. Math. Sci. 50 (2003), 3189–3194. Zbl1037.11002MR2012641
10. M. Z. Garaev and I. E. Shparlinski, The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes. Int. Math. Res. Not. 39 (2005), no. 39, 2391–2408. Zbl1162.11378MR2181356
11. V. C. García, F. Luca and V. J. Mejía, On sums of Fibonacci numbers modulo $p$. Bull. Aust. Math. Soc. 83 (2011), 413–419. Zbl1238.11011MR2794527
12. A. A. Glibichuk, Combinatorial properties of sets of residues modulo a prime and the Erdős–Graham problem. Mat. Zametki. 79 (2006), no. 3, 384–395; English transl., Math. Notes. 79 (2006). no. 3–4, 356–365. Zbl1129.11004MR2251362
13. B. Green and S. V. Konyagin, On the Littlewood problem modulo a prime. Canad. J. Math. 61 (2009), no. 1, 141–164. Zbl1232.11013MR2488453
14. A. A. Karatsuba, An estimate of the $L_1$-norm of an expontential sum. Math. Notes 64 (1998), no. 3, 401–404. Zbl0924.11065MR1680181
15. S. V. Konyagin, On a problem of Littlewood. Izv. Acad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.] 45 (1981), no. 2, 243–265. Zbl0493.42004MR616222
16. S. V. Konyagin, An estimate of the $L_1$-norm of an exponential sum. The Theory of Approximations of Functions and Operators. Abstracts of Papers of the International Conference Dedicated to Stechkin’s 80th Anniversary [in Russian]. Ekaterinburg, 2000, pp. 88-89. 
17. O. C. McGehee, L. Pigno and B. Smith, Hardy’s inequality and the $L^1$ norm of exponential sums. Ann. of Math. (2) 113 (1981), no. 3, 613–618. Zbl0473.42001MR621019
18. F. Pappalardi, On the order of finitely generated subgroups of ${\mathbb{Q}}^{*}(modp)$ and divisors of $p-1$. J. Number Theory 57 (1996), 207–222. Zbl0847.11049MR1382747
19. A. Sárközy, On sums and products of residues modulo $p$. Acta Arith. 118 (2005), no. 4, 403–409. Zbl1078.11011MR2165553
20. T. Schoen and I. Shkredov, Additive properties of multiplicative subgroups of ${𝔽}_p$. Quart. J. Math. 63 (2012), no. 3, 713–722. Zbl1271.11014MR2967172
21. I. E. Shplarlinski, On a question of Erdős and Graham. Arch. Math. 78 (2002), 445–448. Zbl1034.11012MR1921733