Sum-Difference Exponents for Boundedly Many Slopes
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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.
Context
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
The arithmetic Kakeya route (Katz-Tao 2002) is a known pathway to the full Kakeya conjecture, one of harmonic analysis's central open problems.
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"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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