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"?
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.
protected by Todd Trimble♦ Feb 15 at 2:12
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