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Sum-Difference Exponents for Boundedly Many Slopes

Analysis · posed by Nets Katz, Terence Tao (arithmetic Kakeya program), 2002 · partial

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Statement

The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate controlled by a new notion of rational complexity - mapping where the conjectured route cannot succeed.

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  1. proof attempt · #1

    Terence Tao, using AlphaEvolve

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    "Inspired by numerical explorations from the tool AlphaEvolve" - the tool's experiments pointed at the bounded-slope regime and its convergence behaviour; the theorems are Tao's.

    Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.

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