Pseudocontinuation and properties of analytic functions on boundary sets of positive measure

The following analog of Fabry's theorem is proved:
If a function $\mathcal F$ analytic in the polydise $\mathbb D^n$ has a very lacunary Taylor series and coincides (in a certain sense) on a set of positive measure in $\mathbb T^n$ with a function analytic in a sufficiently large subset of $(\mathbb C\setminus\bar{\mathbb D})^n$ then $\mathcal F$ is analytic in the polydisc $(r\mathbb D)^n$ with $r>1$.
This implies that a nonconstant analytic function in the ball $B\subset\mathbb C^n$ with very lacunary Taylor series cannot have nontangential boundary walues with constant modulus or zero real part on a set of positive measure.