Variations on Salem–Zygmund results for random trigonometric polynomials: application to almost sure nodal asymptotics

On some probability space (Ω,F,P), we consider two independent sequences (ak)k≥1 and (bk)k≥1 of i.i.d. random variables that are centered with unit variance which admit a moment strictly higher than two. We then the associated trigonometric polynomial fn(t):=1 n ∑k=1nakcos(kt)+bksin(kt), t∈R. In their seminal work, for Rademacher coefficients, Salem Zygmund showed P almost surely: ∀t∈R,1 2π∫02πexpitf n(x)dx→n→∞e−t2 2. other words, if X denotes an variable uniformly distributed over [0,2π], surely, under law X, fn(X) converges in distribution to standard Gaussian variable. this paper, revisit above Salem–Zygmund result from different perspectives. Namely, establish convergence rate adequate metric via Stein’s method, prove functional counterpart CLT, extend it more general distributions also actually holds total variation. As application, case where coefficients have symmetric order 4, show that, any interval [a,b]⊂[0,2π], number real zeros N(fn,[a,b]) fn [a,b] satisfies universal asymptotics n→n→+∞(b−a) π 3.