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### FIELD Math: CMC December 03 (Tue), 2019 16:30-18:00 8309 Omri Sarig Koh, Kyewon Weizmann Institute of Science Temporal distributional limit theorems for irrational rotations Let f be a Riemann integrable function with zero mean on the unit circle S^1, \alpha an irrational angle, and x some point in S^1. We are interested in the asymptotic behavior of the Birkhoff sums S_n(x):=f(x)+f(x+\alpha)+...+f(x+n\alpha).Weyl's equi-distribution theorem says that S_n(x)=o(n) uniformly in x. A temporal distribution theorem is a scaling limit for the distributions (histograms) of the numbers (S_1(x),S_2(x),...,S_n(x)) for x,f FIXED and n tending to infinity. Such results sometimes hold for special choices of functions f. If and when they hold, they give information on o(n) term in Weyl's theorem, and expose qualitative and quantitative differences between the asymptotic behavior of S_n(x) for different x --- even though the map is uniquely ergodic.I will survey results by J. Beck, Bromberg & Ulcigrai, and joint work with D. Dolgopyat on temporal distributional limit theorems for irrational rotations, and will try to make the talk accessible to a wide audience. S_n(x):=f(x)+f(x+\alpha)+...+f(x+n\alpha).Weyl's equi-distribution theorem says that S_n(x)=o(n) uniformly in x. A temporal distribution theorem is a scaling limit for the distributions (histograms) of the numbers (S_1(x),S_2(x),...,S_n(x)) for x,f FIXED and n tending to infinity. Such results sometimes hold for special choices of functions f. If and when they hold, they give information on o(n) term in Weyl's theorem, and expose qualitative and quantitative differences between the asymptotic behavior of S_n(x) for different x --- even though the map is uniquely ergodic I will survey results by J. Beck, Bromberg & Ulcigrai, and joint work with D. Dolgopyat on temporal distributional limit theorems for irrational rotations, and will try to make the talk accessible to a wide audience. ~