Depth-3 circuits for inner product

What is the Σ32-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2αnn for some αn=Ω(1). We show that the limiting constant α≔limsupαn satisfies0.847...≤α≤0.965.... Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that α∈[0.5,1]. To obtain our improved bounds, we analyse a covering LP that captures the Σ32-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.