We prove that the sign-rank of the $k$-Hamming Distance matrix on $n$ bits is $2^{O(k)}$, independent of the number of bits $n$. This strongly refutes the conjecture of Hatami, Hatami, Pires, Tao, and Zhao (RANDOM 2022), and Hatami, Hosseini, and Meng (STOC 2023), repeated in several other papers, that the ... more >>>

I give a very simple, apparently new proof of a tight communication lower bound for pointer chasing.

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Aaronson (STOC 2010) conjectured that almost $k$-wise independence fools constant-depth circuits; he called this the generalised Linial-Nisan conjecture. Aaronson himself later found a counterexample for depth-3 circuits. We give here an improved counterexample for depth-2 circuits (DNFs). This shows, for instance, that Bazzi's celebrated result ($k$-wise independence fools DNFs) cannot ... more >>>