Before revision:  We show that for any n^{-2}\le\epsilon\le 1, if  for some r \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = \tau\} \ge 2^{-\epsilon n}\cdot 2^{n}\]  \in R, then \[\#\{\langle \xi, w \rangle : \xi \in \{0,1\}^{n}\} \le 2^{O(\epsilonn)}.
After revision:  We show that for any n^{-2}\le\epsilon\le 1, if  \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = \tau\} \ge 2^{-\epsilon n}\cdot 2^{n}\] \in R, then \[\#\{\langle \xi, w \rangle : \xi \in \{0,1\}^{n}\} \le 2^{O(\epsilonn)}.