On cubic lacunary Fourier series

Abstract:

For $2<\beta <4$, we analyze the behavior, near the rational points $x=p\pi /q$, of $\sum ^\infty _{n=1}n^{-\beta }\exp (ixn^{3})$, considered as a function of $x$. We expand this series into a constant term, a term on the order of $(x-p\pi /q)^{(\beta -1)/3}$, a term linear in $x-p\pi /q$, a “chirp" term on the order of $(x-p\pi /q)^{(2\beta -1)/4}$, and an error term on the order of $(x-p\pi /q)^{\beta /2}$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $\beta \le (\sqrt {97}-1)/4=2.212\dots$, both the real and imaginary parts of the cubic series are differentiable almost nowhere.