Can we compute an $n$-bit threshold gate by polynomial size (unbounded fan-in) circuits of depth $\frac{\lg n}{\lg \lg n}$? Alternatively, can we count the number of 1s in the input bits using these circuits?
Is $\mathsf{TC^0} \subseteq \mathsf{AltTime}(O(\frac{\lg n}{\lg \lg n}), O(\lg n))$?
Note that $\mathsf{TC^0} \subseteq \mathsf{NC^1} = \mathsf{ALogTime} = \mathsf{AltTime}(O(\lg n), O(\lg n))$. So the question is essentially asking if we can save a $\lg \lg n$ factor in the depth of circuits when computing threshold gates.
Edit:
As Kristoffer wrote in his answer we can save a $\lg \lg n$ factor. But can we save a little bit more? Can we replace $O(\frac{\lg n}{\lg \lg n})$ with $o(\frac{\lg n}{\lg \lg n})$?
It seems to me that the layered brute-force trick doesn't work for saving even $2 \lg \lg n$ (more generally any function in $\lg \lg n + \omega(1)$).