exponential variation
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2019 ◽  
Vol 21 (1) ◽  
pp. 187-197
Author(s):  
Amit Sharma ◽  
Naveen Mani ◽  
Reeta Bhardwaj


2018 ◽  
Vol 40 (7) ◽  
pp. 1729-1737
Author(s):  
M. ABRAMOFF ◽  
D. BEREND

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.



2017 ◽  
Vol 24 (2) ◽  
pp. 637-647 ◽  
Author(s):  
S. S. P. M. Isa ◽  
N. M. Arifin ◽  
R. Nazar ◽  
N. Bachok ◽  
F. M. Ali ◽  
...  


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