  A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$
is supported on a set of Hausdorff dimension strictly less than $d$
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of $\Z
^d$, $d\geq 2$. By refining the argument, we prove that for all $\b>0$ there
exists $\rho (d,\b)<d$ and $N(d,\b)$, such that for any $n>N(d,\b)$, any $x \in
\Z^d$, and any $A\subset \{1,..., n\}^d$ $$ | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, $$ where $\nu_{A,x} (y)$ denotes the
probability that $y$ is the first entrance point of the simple random walk
starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as $\b \to
\infty$.
