Sum-Difference Exponents for Boundedly Many Slopes
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.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Arithmetic combinatorics
- Posed by
- Nets Katz, Terence Tao (arithmetic Kakeya program)
- Year posed
- 2002
- Years open
- 23y
- Solved
- 2025-11-19
- Model
- AlphaEvolve
- Vendor
- Google DeepMind
- Collaborators
- Terence Tao
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
What the AI did
"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.