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A polynomial-exponential variation of Furstenberg’s $\times 2\times 3$ theorem

Part of: Topological dynamics Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  19 December 2018

M. ABRAMOFF  and
D. BEREND Open the ORCID record for D. BEREND [Opens in a new window]
M. ABRAMOFF
Affiliation:
Department of Mathematics, Ben-Gurion University, Beer Sheva 84105, Israel email abramoff@post.bgu.ac.il
D. BEREND
Affiliation:
Departments of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel email berend@math.bgu.ac.il
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Abstract

Furstenberg’s $\times 2\times 3$ theorem asserts that the double sequence $(2^{m}3^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ is dense modulo one for every irrational $\unicode[STIX]{x1D6FC}$. The same holds with $2$ and $3$ replaced by any two multiplicatively independent integers. Here we obtain the same result for the sequences $((\begin{smallmatrix}m+n\\ d\end{smallmatrix})a^{m}b^{n}\unicode[STIX]{x1D6FC})_{m,n\geq 1}$ for any non-negative integer $d$ and irrational $\unicode[STIX]{x1D6FC}$, and for the sequence $(P(m)a^{m}b^{n})_{m,n\geq 1}$, where $P$ is any polynomial with at least one irrational coefficient. Similarly to Furstenberg’s theorem, both results are obtained by considering appropriate dynamical systems.

Keywords

density modulo 1topological dynamicsFurstenberg’s diophantine theorem

MSC classification

Primary: 11J71: Distribution modulo one 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1729 - 1737
Copyright
© Cambridge University Press, 2018

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