2022/08/21 by Garitsis, Efstathios Konstantinos Chrontsios, Hildebrand, AJ
#11M26 #26A16 #28A80 #42A16 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2208.09806
Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form F(t)=∑n=1^∞ f(n)e2πi nt/n, for a large class of coefficient functions f. Our main result states that if, for some constants C and α with 0<α<1, we have |∑1≤ n≤ xf(n)e2πi nt|≤ C xα uniformly in x≥ 1 and t∈ ℝ, then the series F(t) is Hölder continuous with exponent 1-α, and the graph of |F(t)| on the interval [0,1] has box-counting dimension ≤ 1+α. As applications we recover the best-possible uniform Hölder exponents for the Weierstrass functions ∑k=1^∞ akcos(2πbk t) and the Riemann function ∑n=1^∞ sin(πn2 t)/n2. Moreoever, under the assumption of the Generalized Riemann Hypothesis, we obtain nontrivial bounds for Hölder exponents and dimensions associated with series of the form ∑n=1^∞ μ(n)e2πi nkt/nk, where μ is the Möbius function.