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A polynomial-exponential variation of Furstenberg’s $\times 2\times 3$ theorem
Published online by Cambridge University Press: 19 December 2018
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.
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- © Cambridge University Press, 2018
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