Thematic Program on

Dynamical Systems
School of Mathematics, IPM,
February - May, 2017

 School of Mathematics

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 Mini Course
 Speaker: Omid Hatami   (IPM, Tehran) Title: A quick introduction to Ergodic Theoretic and Analytic aspects of Additive Combinatorics Date & Time: Tuesday, April 18, 2017, 9:30--10:50 Tuesday, April 18, 2017, 11:10--12:30 Thursday, April 20, 2017, 9:30--10:50 Thursday, April 20, 2017, 11:10--12:30 Location: Lecture Hall 2, IPM Niavaran Building, Niavaran Square, Tehran

 Description: In 1936, Erdős and Turán [ET] conjectured that every set of integers A with positive natural density contains a $k$-term arithmetic progression for every $k$. In 1953 Roth [R] proved Erdős-Turán's conjecture for $k = 3$. Later Szemerédi gave a proof first for $k = 4$ [Sz1] and then for general $k$ [Sz24]. In 1977 Furstenberg [F] gave a proof by ergodic theoretic techniques and in 1998 Gowers [G] gave a proof for Szemerédi's Theorem using higher Fourier analysis. In this course I will sketch ergodic, depending on the amount of time available I will review some of the history of the subject. I will also sketch the analytic proof of Szemerédi theorem as well as the ergodic proof and I will try to describe their connection.

 References: [ET] P. Erdős and P. Turán, On some sequences of integers, Journal of the London Mathematical Society. 11 (4), (1936), 261-264. [R] K. F. Roth, On certain sets of integers, Journal of the London Mathematical Society. 28 (1), (1953), 104-109. [Sz1] E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hung. 20, (1969), 89-104. [Sz2] E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arithmetica 27, (1975), 199-245. [F] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. D'Analyse Math. 31, (1977), 204-256. [G] T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (3), (2001), 465--588.

 School of Mathematics, IPM - Institute for Research in Fundamental Sciences Niavaran Building, Niavaran Square, Tehran, Iran Tel: +98 21 222 90 928 Email: gt@ipm.ir