MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are "easier"?

share|cite|improve this question
You might find Tao's blog post on dyadic models relevant: – Mark Lewko Jul 26 '13 at 9:17
Could someone with high rep protect this question? Otherwise I think Sow will keep showing up . . . – Noah Schweber Feb 15 at 1:01
up vote 23 down vote accepted

The Riemann hypothesis is proved over function fields (like the fraction field of F_q[t]), not finite fields, and the "real version" is a question about the integers. Kakeya is proved over finite fields, and the "real version" is a question about, well, the reals. So the situations are quite different. I'd say that the truth of RH over function fields really does make me feel more confidence that RH is true over number fields, because the analogy between function fields and number fields is in many ways a very close one. On the other hand, the truth of Kakeya over finite fields does not tell me very much about the truth of the real Kakeya problem. For one thing, what's true over finite fields -- that a Kakeya set has measure bounded away from 0 -- is totally false over the real numbers! (See: Besicovich set.) An intermediate multiple-scale case is that of a power series ring over a finite field -- in this case, measure-0 Kakeya sets exist by a theorem of Dummit and Hablicsek, but we don't know the answer to the Kakeya problem, i.e. we don't know whether a Kakeya set over F_q[[t]] has full dimension.

One reason (but not the only reason) the Kakeya problem is easier over finite fields than it is over the real numbers is that finite fields only have one scale, while the real numbers have multiple scales.

One reason (but not the only reason) the Riemann Hypothesis is easier over function fields than it is over number fields is that the zeta function in the function field case has an interpretation in terms of the etale cohomology of a variety over a finite field. In the number field case there is no such interpretation at present, despite the best efforts of the F_un-ologists.

share|cite|improve this answer
excellent answer. – Koushik Aug 7 '13 at 17:32

protected by Todd Trimble Feb 15 at 2:12

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.