Sets of convergence to zero of lacunary trigonometric series (Q752388)

Let \(E\subset [0,2\pi)\) be a set such that any lacunary trigonometric series (in Hadamard's sense) \[ a_ 0/2+\sum^{\infty}_{k=1}a_ k \cos n_ kx+b_ k \sin n_ kx, \] where \(n_{k+1}/n_ k\geq q>1\) \((k=1,2,...)\), which converges to zero on E, is the identically zero series, i.e., \(a_ 0=a_ k=b_ k=0\) \((k=1,2,...)\). The set [0,2\(\pi\))\(\setminus E\) is called a set of uniqueness for lacunary series. We consider the question: Is a set of uniqueness necessarily of the cardinality of the continuum \({\mathfrak c}?\) It turns out that this problem is undecidable in Zermelo-Fraenkel set theory.