2025/08/04 by Steinerberger, Stefan
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.02489
Let x ∈ ℝ be arbitrary and consider the `greedy' approximation of x by signed harmonic sums: given an = ∑k ≤ n εk/k with εk ∈ \-1,1\, we set εn+1 = 1 if an ≤ x and εn+1 = -1 otherwise. Bettin-Molteni-Sanna showed (Adv. Math. 2020) that this procedure has remarkable approximation properties: for almost all x ∈ ℝ one has superpolynomial convergence in the sense that for every k ∈ ℕ there are infinitely many n ∈ ℕ with |an - x| ≤ n-k. We extend this result from ± 1 ± 1/2 ± 1/3 … ± 1/n to moment sequences, i.e. sequences defined as the moments of a measure μ supported on [0,1].