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The Matrix Spencer Conjecture for Finite Groups

Probability & statistics · posed by Nikhil Bansal, Haotian Jiang, Raghu Meka, 2022 · solved

1 attempt

Statement

The group version of the Matrix Spencer conjecture holds: for every finite group GG there are signs ε{±1}G\varepsilon \in \{\pm 1\}^G with gGεgρ(g)CG\left\|\sum_{g \in G} \varepsilon_g \rho(g)\right\| \le C\sqrt{|G|}, where ρ\rho is the left regular representation and CC is universal.

Context

the group case; the full Matrix Spencer conjecture remains open

The Matrix Spencer conjecture is the matrix analogue of Spencer's six standard deviations theorem and one of the best known open problems in discrepancy theory, with a decade of partial results behind it.

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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptChatGPT Pro 5.5, Claude Opus 4.7 and 4.8 with Afonso S. Bandeira, Helmut Bolcskei ·
    AI involvement
    ai co developed
    a person and a model developed the result together.
    models
    ChatGPT Pro 5.5, Claude Opus 4.7 and 4.8
    people
    Afonso S. Bandeira, Helmut Bolcskei

    The acknowledgement says modern AI tools were used throughout, and singles out one of the key arguments as the one that broke the problem open, crediting it to that use.

    the group case; the full Matrix Spencer conjecture remains open

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