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I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a topic of Diophantine Approximation and there are lots of research relating Ergodic Theory and Diophantine Approximation. I even took a course and studied Ergodic Theory book by Walters and Ergodic Theory with a View towards Number Theory book by Einsiedlerhoping. Then I searched lots of papers but there is no sign of proving irrationality of specific constants by methods of Ergodic Theory. On the other hand, there are famous unsolved problems in Number Theory that are solved recently (fully or almost fully) by Ergodic Theory like the Littlewood's conjecture, Green-Tao theorem, Erdos Discrepancy Problem, etc. There are lots of papers also relating Ergodic Theory and Diophantine Approximation but nothing about proving irrationality of numbers.

So why thereMy question is no sign of evenwould you please cite a very little about Euler-Mascheroni Constant, etc. inpaper regarding proving irrationality of numbers through Ergodic Theory methods?

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a topic of Diophantine Approximation and there are lots of research relating Ergodic Theory and Diophantine Approximation. I even took a course and studied Ergodic Theory book by Walters and Ergodic Theory with a View towards Number Theory book by Einsiedlerhoping. Then I searched lots of papers but there is no sign of proving irrationality of specific constants by methods of Ergodic Theory. On the other hand, there are famous unsolved problems in Number Theory that are solved recently (fully or almost fully) by Ergodic Theory like the Littlewood's conjecture, Green-Tao theorem, Erdos Discrepancy Problem, etc. There are lots of papers also relating Ergodic Theory and Diophantine Approximation but nothing about proving irrationality of numbers.

So why there is no sign of even a very little about Euler-Mascheroni Constant, etc. in Ergodic Theory?

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a topic of Diophantine Approximation and there are lots of research relating Ergodic Theory and Diophantine Approximation. I even took a course and studied Ergodic Theory book by Walters and Ergodic Theory with a View towards Number Theory book by Einsiedlerhoping. Then I searched lots of papers but there is no sign of proving irrationality of specific constants by methods of Ergodic Theory. On the other hand, there are famous unsolved problems in Number Theory that are solved recently (fully or almost fully) by Ergodic Theory like the Littlewood's conjecture, Green-Tao theorem, Erdos Discrepancy Problem, etc. There are lots of papers also relating Ergodic Theory and Diophantine Approximation but nothing about proving irrationality of numbers.

My question is would you please cite a paper regarding proving irrationality of numbers through Ergodic Theory methods?

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user288447
user288447

Ergodic Theory and Euler-Mascheroni Constant

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a topic of Diophantine Approximation and there are lots of research relating Ergodic Theory and Diophantine Approximation. I even took a course and studied Ergodic Theory book by Walters and Ergodic Theory with a View towards Number Theory book by Einsiedlerhoping. Then I searched lots of papers but there is no sign of proving irrationality of specific constants by methods of Ergodic Theory. On the other hand, there are famous unsolved problems in Number Theory that are solved recently (fully or almost fully) by Ergodic Theory like the Littlewood's conjecture, Green-Tao theorem, Erdos Discrepancy Problem, etc. There are lots of papers also relating Ergodic Theory and Diophantine Approximation but nothing about proving irrationality of numbers.

So why there is no sign of even a very little about Euler-Mascheroni Constant, etc. in Ergodic Theory?