On Extremal Properties of k-CNF: Capturing Threshold Functions

We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula that only accepts assignments of weight at least $t$? Among others, we provide the following results: - While an optimal solution is known for $t\le n\mathrm{/}k$, the problem remains open for $t>n\mathrm{/}k$. We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t=n-k$, the problem is exactly the Turán problem. - For $t=\alpha n$ with constant $\alpha$, we provide a construction and show its optimality for $2$-CNF. Optimality of the construction for $k>2$ would give improved lower bounds for depth-$3$ circuits.