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Irrational dilations of Pascal's triangle

Part of:Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

D. Berend,
M. D. Boshernitzan and
G. Kolsenik
D. Berend
Affiliation:
Departments of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
M. D. Boshernitzan
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77251, USA.
G. Kolsenik
Affiliation:
Department of Mathematics and Computer Science, California State University, Los Angeles, CA 90032, USA.
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Extract

Let (bn) be a sequence of integers, obtained by traversing the rows of Pascal's triangle, as follows: start from the element at the top of the triangle, and at each stage continue from the current element to one of the elements at the next row, either the one immediately to the left of the current element or the one immediately to its right. Consider the distribution of the sequence (bnα) modulo 1 for an irrational α. The main results show that this sequence “often” fails to be uniformly distributed modulo 1, and provide answers to some questions raised by Adams and Petersen.

MSC classification

Secondary:11J71: Distribution modulo one

Information

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001, pp. 159 - 168
Copyright
Copyright © University College London 2001

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References

1.Adams, T. M. and Petersen, K. E.. Binomial-coefficient multiples of irrationals. Monats. Math. 125 (1998), 269–278.CrossRefGoogle Scholar
2.Furstenberg, H.. Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation. Math. Syst. Th. 1 (1967), 1–49.Google Scholar
3.Goetgheluck, P.. Infinite families of solutions of the equation . Math. Comp. 67 (1998), 1727–1733.Google Scholar
4.Karacuba, A. A.. Estimates for trigonometric sums by Vinogradov's method, and some applications. Proc. Steklov Inst. Math. 112, Amer. Math. Soc. (Providence, RI, 1973), 251–265.Google Scholar
5.Korobov, N. M.. Exponential Sums and their Applications. Kluwer (Dordrecht, The Netherlands, 1992).Google Scholar
6.Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences. Wiley-Interscience (New York, 1974).Google Scholar
7.Lucas, E.. Théorie des fonctions numériques simplement périodiques. Amer. J. Math. 1 (1878), 184–240, 289-321.Google Scholar
8.Mathan, B. de. Numbers contravening a condition in density modulo 1. Ada Math. Acad. Sci. Hungar. 36 (1980), 237–241.Google Scholar
9.Pollington, A. D.. On the density of sequence {nkξ}. Illinois J. Math. 23 (1979), 511–515.CrossRefGoogle Scholar
10.Petersen, K. E. and Schmidt, K.. Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349 (1997), 2775–2811.CrossRefGoogle Scholar
11.Vershik, A. M.. Descriptions of invariant measures for actions of infinite groups. Dokl. Akad. Nauk SSSR 218 (1974), 749-752; Sower Math. Dokl. 15 (1974), 1396-1400.Google Scholar