Tight Correlation Bounds for Circuits Between AC0 and TC0

We initiate the study of generalized AC0 circuits comprised of negations and arbitrary unbounded fan-in gates that only need to be constant over inputs of Hamming weight $\ge k$, which we denote GC0$(k)$. The gate set of this class includes biased LTFs like the $k$-$OR$ (output $1$ iff $\ge k$ bits are 1) and $k$-$AND$ (output $0$ iff $\ge k$ bits are 0), and thus can be seen as an interpolation between AC0 and TC0. We establish a tight multi-switching lemma for GC0$(k)$ circuits, which bounds the probability that several depth-2 GC0$(k)$ circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-$d$ size-$s$ AC0 circuits lifts to depth-$d$ size-$s^{.99}$ GC0$(.01\log s)$ circuits with no loss in parameters (other than hidden constants). Our result has the following applications: 1.Size-$2^{\Omega(n^{1/d})}$ depth-$d$ GC0$(\Omega(n^{1/d}))$ circuits do not correlate with parity (extending a result of H{\aa}stad (SICOMP, 2014)). 2. Size-$n^{\Omega(\log n)}$ GC0$(\Omega(\log^2 n))$ circuits with $n^{.249}$ arbitrary threshold gates or $n^{.499}$ arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)). 3. There is a seed length $O((\log m)^{d-1}\log(m/\varepsilon)\log\log(m))$ pseudorandom generator against size-$m$ depth-$d$ GC0$(\log m)$ circuits, matching the AC0 lower bound of H{\aa}stad stad up to a $\log\log m$ factor (extending a result of Lyu (CCC, 2022)). 4. Size-$m$ GC0$(\log m)$ circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).