Let w = (w_1,\dots, w_n) \in R^{n}. We show that for any n^{-2}\le\epsilon\le 1, if %DIFDELCMD < \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = r\} \ge 2^{-\epsilon n}\cdot 2^{n}\] %%% for some r \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = \tau\} \ge 2^{-\epsilon n}\cdot 2^{n}\]  \in R, then \[\#\{\langle \xi, w \rangle : \xi \in \{0,1\}^{n}\} \le 2^{O(\epsilonn)}.\] This exponentially improves  a recent result of Nederlof, Pawlewicz, Swennenhuis, and W\k{e}grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.
Let w = (w_1,\dots, w_n) \in R^{n}. We show that for any n^{-2}\le\epsilon\le 1, if %DIFDELCMD < \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = r\} \ge 2^{-\epsilon n}\cdot 2^{n}\] %%% \[\#\{\xi \in \{0,1\}^{n}: \langle \xi, w \rangle = \tau\} \ge 2^{-\epsilon n}\cdot 2^{n}\] for some \tau \in R, then \[\#\{\langle \xi, w \rangle : \xi \in \{0,1\}^{n}\} \le 2^{O(\epsilonn)}.\] This exponentially improves the \epsilon dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and W\k{e}grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.
D
for some r
A
for some \tau
A
the \epsilon dependence in