  We consider the Hausdorff dimension of the divergence set on which the
pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-\Delta}} f(x) = f(x)$
fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised
by Barcel\'o, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve
the previous results in higher dimensions $d\ge4$. We also show that a
Strichartz type estimate for $f\to e^{it\sqrt{-\Delta}} f$ with the measure $
dt\,d\mu(x)$ is essentially equivalent to the estimate for the spherical
average of $\widehat \mu$ which has been extensively studied for the Falconer
distance set problem. The equivalence provides shortcuts to the recent results
due to B. Liu and K. Rogers.
