What the study found
The study found that a Kakeya set in a vector space over a finite field supports a probability measure whose Fourier transform is bounded for all non-zero frequencies. The authors also report that this bound is sharp in all dimensions at least 2.
Why the authors say this matters
The authors say this gives a Fourier analytic proof that a Kakeya set in dimension 2 must have size at least the stated lower bound, which they describe as asymptotically sharp. The study also suggests analogous results for sets containing planes in specified orientations.
What the researchers tested
The paper studies Kakeya sets in vector spaces over finite fields using Fourier analysis. It also considers sets containing planes in a given set of orientations.
What worked and what didn't
The authors prove the Fourier transform bound for non-zero frequencies and show that it cannot be improved in dimensions 2 and above. They also obtain an analogous result for sets containing planes in specified orientations.
What to keep in mind
The abstract does not provide the exact numerical bounds, so those details are not included here. It also does not describe limitations beyond the stated scope over finite fields and dimensions at least 2.
Key points
- Kakeya sets over finite fields support a probability measure with a bounded Fourier transform at non-zero frequencies.
- The bound is reported to be sharp in all dimensions at least 2.
- The authors say this yields a Fourier analytic proof of a lower bound on the size of two-dimensional Kakeya sets.
- The abstract also mentions analogous results for sets containing planes in specified orientations.
Disclosure
- Research title:
- Fourier bounds for Kakeya sets in finite fields
- Authors:
- Jonathan M. Fraser
- Institutions:
- University of St Andrews
- Publication date:
- 2026-04-20
- DOI:
- 10.1112/blms.70367
- OpenAlex record:
- View
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