A polynomial-exponential variation of Furstenberg’s theorem
| Autor: | M. Abramoff, D. Berend |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
medicine.medical_specialty
Polynomial Pure mathematics Dynamical systems theory Applied Mathematics General Mathematics 010102 general mathematics Topological dynamics 01 natural sciences Exponential function Variation (linguistics) Irrational number 0103 physical sciences medicine 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | Ergodic Theory and Dynamical Systems. 40:1729-1737 |
| ISSN: | 1469-4417 0143-3857 |
| DOI: | 10.1017/etds.2018.132 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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