In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function.
Specifically, we prove that for every non-linear and symmetric $f:\{0,1\}^{k} \to \{0,1\}$ there exists a set $\emptyset\neq S\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S) \neq 0$, where ... more >>>

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>

We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, small-bias spaces, 0/1 halfspaces, and 0/1 modular sums. A function $f:[m]^n \rightarrow \{0,1\}^n$ is an $(m,n)$-combinatorial shape if there exist sets $A_1,\ldots,A_n \subseteq [m]$ and a symmetric function $h:\{0,1\}^n \rightarrow \{0,1\}$ such that $f(x_1,\ldots,x_n) = h(1_{A_1} (x_1),\ldots,1_{A_n}(x_n))$. Our ... more >>>