Visible points on modular exponential curves

• Autorzy: Chan T. H. , Shparlinski I. E.
• Języki publikacji: EN

Abstrakty

EN

We obtain an asymptotic formula for the number of visible points (x,y). that is, with gcd(x, y) = 1, which lie in the box [1, U] x [1,V] and also belong to the exponential modular curves y ≡ agx (mod p). Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

Słowa kluczowe

EN

exponential congruences; visible points

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: Vol. 58, no 1
• Strony: 17--22
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Chan T. H.

autor

Shparlinski I. E.

• Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A.,

Bibliografia

• [1] J. Bourgain and M. Z. Garaev, On a variant of sum-product estimates and explicit exponential sum bounds in prime fields, Math. Proc. Cambridge Philos. Soc. 146 (2008), 1-21.
• [2] T. H. Chan and I. E. Shparlinski, On the concentration of points on modular hyperbolas and exponential curves, Acta Arith. 142 (2010), 59-66.
• [3] C. Cobeli, S. Gonek and A. Zaharescu, On the distribution of small powers of a primitive root, J. Number Theory 88 (2001), 49-58.
• [4] M. Z. Garaev, On the logarithmic factor in error term estimates in certain additive congruence problems, Acta Arith. 124 (2006), 27-39.
• [5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, The Clarendon Press, Oxford Univ. Press, New York, 1979.
• [6] S. V. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and Their Applications, Cambridge Univ. Press, Cambridge, 1999.
• [7] N. M. Korobov, On the distribution of digits in periodic fractions, Mat. Sb. 89 (1972), 654-670 (in Russian).
• [8] H. L. Montgomery, Distribution of small powers of a primitive root, in: Advances in Number Theory, Clarendon Press, Oxford, 1993, 137-149.
• [9] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), 957-1041.
• [10] I. E. Shparlinski, Primitive points on a modular hyperbola, Bull. Polish Acad. Sci. Math. 54 (2006), 193-200.
• [11] I. E. Shparlinski and J. F. Voloch, Visible points on curves over finite fields, ibid. 55 (2007), 193-199.
• [12] I. E. Shparlinski and A. Winterhof, Visible points on multidimensional modular hyperbolas, J. Number Theory 128 (2008), 2695-2703.