2022/02/17 by Oleksiy Klurman, Klurman, Oleksiy, Ilya D. Shkredov +3 · 4 citations
Mathematics · #Stochastic processes and statistical mechanics #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · doi:10.48550/arxiv.2202.08767
We show that for a Steinhaus random multiplicative function f:ℕ→\mathbbD and any polynomial P(x)∈ℤ[x] of deg P≥ 2 which is not of the form w(x+c)d for some w∈ ℤ, c∈ ℚ, we have (1)/(√(x))∑n≤ x f(P(n)) \xrightarrowd CN(0,1), where CN(0,1) is the standard complex Gaussian distribution with mean 0 and variance 1. This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of x≥ 1, such that |∑n≤ x f(P(n))| ≫deg P √(x) (log log x)1/2, for any polynomial P(x)∈ℤ[x] with deg P≥ 2, which is not a product of linear factors (over ℚ). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase P(n)=n, where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be O(√(x)(log log x)(1)/(4)+ε) for any ε>0.