We prove that the Or function on $n$ bits can be point-wise approximated with error $\eps$ by a polynomial of degree $O(k)$ and weight $2^{O(n \log (1/\eps)/k)}$, for any $k \geq \sqrt{n \log 1/\eps}$. This result is tight for all $k$. Previous results were either not tight or had $\eps ... more >>>

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in [Atserias-Müller,2019] asks whether a certain natural propositional encoding of the above statement is ... more >>>

The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graph’s vertex set. A graph property $\mathcal{P}$ is said ... more >>>