Examples of problems where considering “discrete analogues” has provided insight or led to a solution of the original problem

The Kakeya conjecture posits that any Kakeya set in $$\mathbb{R}^n$$ has dimension $$n$$.

A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir in 2008.

My understanding is that the Finite Field Kakeya Conjecture was proposed at the end of the twentieth century with the hope that it would lead to methods that could be applied to the original Kakeya conjecture. However, it seems that in this case the approach used for resolving the discrete analogue is not easily applied to the original problem, so that the Kakeya conjecture still remains open.

My question is asking for examples of problems where discretizing "succeeded."

Question: Are there examples of math problems where looking at a finite or discrete variant of the initial statement did lead to a solution (or if not a complete solution, at least significant progress) of the original problem? If so, what are they?

Edit:

As commenters pointed out, this question is similar to the one here which requests examples where a discrete version of a theory was developed before the continuous version of the same theory.

My question is different from that previous question, in that I am not interested in cases where discrete problems predated continuous problems. Instead, I'd like to learn about instances where a continuous problem was already proposed, and studying a discrete version of that problem helped inform a solution to the continuous variant.

• Does the exhaustion method qualify? – Sylvain JULIEN Apr 20 at 16:50
• Possible duplicate of When has discrete understanding preceded continuous? – Igor Pak Apr 21 at 1:37
• It's also worth noting that Thomas Wolff, who proposed the Finite Field Kakeya conjecture, had already by that time (1999) initiated a rather successful program of applying ideas from incidence geometry (a discrete field) to problems of Fourier analysis. (As a starting point see e.g. these notes of Larry Guth: math.mit.edu/~lguth/Namboodiri/namboodiri3.pdf). – Sam Hopkins Apr 21 at 2:52
• @SamHopkins I don't think that's quite the same. KS was shown to be equivalent to a discrete problem (with hindsight this is natural, KS has ultrafilters in the background). The OP is asking about discrete analogues of a continuous problem, whereas the paving problem and Nik Weaver's work are really about showing the continuous problem is equivalent to a combinatorial question – Yemon Choi Apr 21 at 3:20
• While Dvir's arguments have not yet led to a solution of the continuous Kakeya conjecture, the polynomial interpolation method of Dvir did inspire what is now known as the polynomial partitioning method, which did establish several variants of the continuous Kakeya problem, starting with this paper of Guth: arxiv.org/abs/0811.2251 – Terry Tao Apr 21 at 16:43

In my paper

Tao, Terence, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dyn. Syst. 28, No. 2, 657-688 (2008). ZBL1181.37004.

I was able to settle a question in ergodic theory (namely, the norm convergence of averages $$\frac{1}{N} \sum_{n=1}^N \int_X T_1^n f_1 \dots T_k^n f_k\ d\mu$$ for $$k$$ commuting measure-preserving transformations $$T_1,\dots,T_k$$ on a probability space $$(X,\mu)$$ and bounded functions $$f_1,\dots,f_k$$), by abandoning all the usual ergodic theory machinery (e.g., characteristic factors), and translating the problem to a purely finitary one without explicit use of limits. This could then be attacked by methods related to graph and hypergraph regularity. The argument was then significantly generalised in

Walsh, Miguel N., Norm convergence of nilpotent ergodic averages, Ann. Math. (2) 175, No. 3, 1667-1688 (2012). ZBL1248.37008.

It should however be pointed out that an alternate, ergodic-theoretic proof of these results was also subsequently given in

Austin, Tim, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dyn. Syst. 30, No. 2, 321-338 (2010). ZBL1206.37003.

Austin, Tim, A proof of Walsh’s convergence theorem using couplings, Int. Math. Res. Not. 2015, No. 15, 6661-6674 (2015). ZBL1372.37012.

I should also mention that there are several papers of Bourgain in which he establishes various ergodic theorems by converting them to quantitative questions in harmonic analysis which are strictly speaking not discrete or finitary, but are amenable to many of the same techniques (in particular, a focus on "hard analysis" estimates) as such discrete problems. A typical such paper is

Bourgain, J., On the pointwise ergodic theorem on $$L^p$$ for arithmetic sets, Isr. J. Math. 61, No. 1, 73-84 (1988). ZBL0642.28011.

As a non-expert, I will tell a story (since I am not qualified to do more):

Once upon a time (1859) there was conjecture about the zeros of a complex function known as the Riemann Hypothesis (RH).

Years later Weil formulated a "discrete version" (in terms of finite fields $$\mathbb{F}_q$$) of this conjecture; the third of his Weil Conjectures (1949). In 1974, Deligne proved this discrete version to much acclaim. Here is a well regarded exposition by Milne.

Years later still, Connes is motivated to explore a potential proof of the original RH along the lines of Deligne's proof by taking a "limit as $$q\to 1$$".

These attempts are probably uncontroversially considered mathematical advances. Some, presumably Connes himself, consider this work an advance in the problem at hand (and so making this hopefully an appropriate response to this MO question). However, there are other opinions, and so this MO post might prove controversial too.

• I thought of things like this, but it is not clear to me that $\mathbb F_q[t]$ is more discrete in a meaningful sense than $\mathbb Z$. In some ways it is more discrete, in others more continuous. – Will Sawin Apr 21 at 16:19
• That said, another example in this vein is Ngo’s Proof of the function field analogue of the fundamental lemma, which led, thanks to earlier work of Waldspurger and others, to a proof of the original fundamental lemma. – Will Sawin Apr 21 at 16:24