Depth-3 Circuits for Inner Product

What is the Σ 23 -circuit complexity (depth 3, bottom-fanin 2) of the 2 n -bit inner product function? The complexity is known to be exponential 2 α n n for some α n = Ω(1). We show that the limiting constant α := lim sup α n satisfies 0 . 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

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