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Lehmer points and visible points on affine varieties over finite fields

Published online by Cambridge University Press:  14 November 2013

KIT-HO MAK  and
ALEXANDRU ZAHARESCU
KIT-HO MAK
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332, U.S.A. e-mail: kmak6@math.gatech.edu
ALEXANDRU ZAHARESCU
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, 1409 W. Green Street, Urbana, Illinois 61801, U.S.A. e-mail: zaharesc@math.uiuc.edu
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Abstract

Let V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 193 - 207
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Alkan, E., Stan, F. and Zaharescu, A.Lehmer k-tuples. Proc. Amer. Math. Soc. 134 (10) (2006), 2807–2815.Google Scholar
[2]Bombieri, E.On exponential sums in finite fields. Amer. J. Math. 88 (1) (1966), 71–105.Google Scholar
[3]Bombieri, E.On exponential sums in finite fields. II. Invent. Math. 47 (1) (1978), 29–39.Google Scholar
[4]Bourgain, J., Cochrane, T., Paulhus, J. and Pinner, C.On the parity of kth powers modulo p. A generalization of a problem of Lehmer. Acta Arith. 147 (2) (2011), 173–203.CrossRefGoogle Scholar
[5]Chan, T. H.Approximating reals by sums of two rationals. J. Number Theory 128 (5) (2008), 1182–1194.Google Scholar
[6]Chan, T. H.Approximating reals by sums of rationals. J. Number Theory 129 (2) (2009), 316–324.Google Scholar
[7]Chan, T. H. and Shparlinski, I.Visible points on modular exponential curves. Bull. Pol. Acad. Sci. Math. 58 (1) (2010), 17–22.Google Scholar
[8]Cilleruelo, J., Garaev, M., Ostafe, A. and Shparlinski, I.On the concentration of points of polynomial maps and applications. To appear in Math. Z. 272 (3-4) (2012), 825–837.Google Scholar
[9]Cobeli, C. and Zaharescu, A.Generalization of a problem of Lehmer. Manuscripta Math. 104 (3) (2001), 301–307.Google Scholar
[10]Deligne, P.La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.Google Scholar
[11]Deligne, P.La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252.Google Scholar
[12]Fouvry, E.Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums. Israel J. Math. 120 (part A) (2000), 81–96.Google Scholar
[13]Fujiwara, M.Distribution of rational points on varieties over finite fields. Mathematika 35 (2) (1988), 155–171.Google Scholar
[14]Guy, R.Unsolved Problems in Number Theory. Problem Books in Mathematics. (Springer-Verlag, New York, second edition, 1994). Unsolved Problems in Intuitive Mathematics, I.Google Scholar
[15]Katz, N.Estimates for “singular” exponential sums. Int. Math. Res. Not. 16 (1999), 875–899.Google Scholar
[16]Laczkovich, M.Discrepancy estimates for sets with small boundary. Studia Sci. Math. Hungar. 30 (1-2) (1995), 105–109.Google Scholar
[17]Lang, S. and Weil, A.Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827.CrossRefGoogle Scholar
[18]Luo, W.Rational points on complete intersections over Fp. Int. Math. Res. Not. 16 (1999), 901–907.Google Scholar
[19]Shparlinski, I.Primitive points on modular hyperbola. Bull. Pol. Acad. Sci. Math. 54 (3-4) (2006), 193–200.Google Scholar
[20]Shparlinski, I. and Voloch, J. F.Visible points on curves over finite fields. Bull. Pol. Acad. Sci. Math. 55 (3) (2007), 193–199.Google Scholar
[21]Shparlinski, I. and Winterhof, A.Visible points on multidimensional modular hyperbolas. J. Number Theory 128 (9) (2008), 2695–2703.Google Scholar
[22]Shparlinski, I. E.On a generalisation of a Lehmer problem. Math. Z. 263 (3) (2009), 619–631.CrossRefGoogle Scholar
[23]Shparlinski, I. E. and Skorobogatov, A. N.Exponential sums and rational points on complete intersections. Mathematika 37 (2) (1990), 201–208.Google Scholar
[24]Skorobogatov, A. N.Exponential sums, the geometry of hyperplane sections, and some Diophantine problems. Israel J. Math. 80 (3) (1992), 359–379.Google Scholar
[25]Weyl, H.On the Volume of Tubes. Amer. J. Math. 61 (2) (1939), 461–472.CrossRefGoogle Scholar
[26]Zhang, W.On a problem of D. H. Lehmer and its generalization. Compositio Math. 86 (3) (1993), 307–316.Google Scholar
[27]Zhang, W.On the difference between a D. H. Lehmer number and its inverse modulo q. Acta Arith. 68 (3) (1994), 255–263.Google Scholar
[28]Zhang, W.A problem of D. H. Lehmer and its generalization. II. Compositio Math. 91 (1) (1994, 47–56.Google Scholar