2023/03/02 by Cătălin Badea, Badea, Catalin, Sophie Grivaux +1
Mathematics · #37A25 (Secondary) #43A25 (Primary) 37A05 #54E52 #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2303.01089
openalex publication_date 2023/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
For each integer n≥ 1, denote by Tn the map x↦ nx\mod 1 from the circle group \mathbbT = ℝ/ℤ into itself. Let p,q≥ 2 be two multiplicatively independent integers. Using Baire Category arguments, we show that generically a Tp-invariant probability measure μ on \mathbbT with no atom has some large Fourier coefficients along the sequence (qn)n≥ 0. In particular, (Tqnμ)n≥ 0 does not converges weak-star to the normalised Lebesgue measure on \mathbbT. This disproves a conjecture of Furstenberg and complements previous results of Johnson and Rudolph. In the spirit of previous work by Meiri and Lindenstrauss-Meiri-Peres, we study generalisations of our main result to certain classes of sequences (cn)n≥ 0 other than the sequences (qn)n≥ 0, and also investigate the multidimensional setting.