Proceedings of the American Mathematical Society

Author(s): G. R. Everest; I. E. Shparlinski
Journal: Proc. Amer. Math. Soc. 127 (1999), 665-675.
MSC (1991): Primary 11B83
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Abstract: We prove an effective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects generalise simultaneously the fundamental examples of linear recurrence sequences and sums of

-units. The proof is based on an effective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also used to obtain a non-effective asymptotic formula counting the norms of these values below a fixed bound.

[1]: G. R. Everest, Counting the values taken by sums of $S$ -units, J. Number Theory 35 (1990), 269-286. MR 91j:11015
[2]: G. R. Everest, On the $p$ -adic integral of an exponential polynomial, Bull. London Math. Soc. 27 (1995), 334-340. MR 96c:11137
[3]: G. R. Everest and I. E. Shparlinski, Divisor sums of generalised exponential polynomials, Bull. Canad. Math. Soc. 53 (1996), 35-46. MR 97g:11021
[4]: J.-H. Evertse, On sums of $S$ -units and linear recurrences, Compositio Math. 53 (1984), 225-244. MR 86c:11045
[5]: S. Lang, Algebraic Number Theory, Addison-Wesley, New York, 1970. MR 44:181
[6]: A. J. van der Poorten and H.-P. Schlickewei, The growth conditions for recurrence sequences, Macquarie Math. Reports 82-0041 (1982).
[7]: A. J. van der Poorten and I. Shparlinski, On the number of zeros of exponential polynomials and related questions, Bull. Austral. Math. Soc. 46 (1992), 401-412.
[8]: H.-P. Schlickewei and W. M. Schmidt, On polynomial-exponential equations, Math. Ann. 296 (1993), 339-361. MR 94e:11032
[9]: -, On polynomial-exponential equations, II (to appear).
[10]: K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math. 111 (1993), 69-76. MR 95c:22011
[11]: W. M. Schmidt, Diophantine Approximation and Diophantine Equations, LNM # 1467, Springer, Berlin, 1991. MR 94f:11059
[12]: T. N. Shorey and R. J. Tijdeman, Exponential Diophantine Equations, CUP, 1986. MR 88h:11002
[13]: I. Shparlinski, On the number of distinct prime divisors of recurrence sequences, Matem. Zametki 42 (1987), 494-507 (Russian).

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11B83

G. R. Everest
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, Norfolk, United Kingdom
Email: g.everest@uea.ac.uk

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ISSN 1088-6826 (e) ISSN 0002-9939 (p)

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