Sum Product Phenomena
Speaker: Terence TaoDate of Talk: May 5, 2026
Upstream link: UCLA Analysis and PDE Seminar
This is a special lecture to supplement a topics course about the proof of the Kakeya conjecture in \(\mathbb{R}^3\).
1. The Erdős-Szeméredi Conjecture
The central phenomenon in question is the following: consider \(A\subseteq \mathbb{R}\) a finite set. Then the set
\[A + A = \left\lbrace a_1 + a_2 : a_1, a_2 \in A \right\rbrace\]has cardinality \(\gtrsim \left\lvert A \right\rvert\) and \(\lesssim \left\lvert A \right\rvert^2\), with the latter being typical of a random finite subset and the former extremising behaviour occurring only when \(A\) is close to an arithmetic progression. One can say the same for \(A\cdot A\), where its cardinality is small insteal for \(A\) close to a geometric progression.
Conjecture 1. Erdős-Szeméredi
For any \(A\subseteq \mathbb{R}\) finite and \(\epsilon >0\),
\[\max \left( \left\lvert A+A \right\rvert, \left\lvert A\cdot A \right\rvert \right) \gtrsim_ \epsilon \left\lvert A \right\rvert ^{2 - \epsilon }.\]Morally speaking, the only way this maximum can be small is if both extremising behaviours happen, i.e. \(A\) is close to a subring of \(\mathbb{R}\). But there are no finite subrings of \(\mathbb{R}\)…
The best known lower bound has an exponent of just barely above \(\frac{1}{3}\).
There is nothing special about \(\mathbb{R}\); one can replace it with any ring or field. For \(\mathbb{C}\), similar results to \(\mathbb{R}\) are known. There is a finite version of the problem too, though one must now exclude \(A\) being “too close” to the whole field. There is minor progress along these lines:
Theorem 2. (Bourgain-Katz-Tao)
For every \(\delta > 0\) and \(A\subseteq \mathbb{F}_p\) with \(p^ \delta \lesssim \left\lvert A \right\rvert \lesssim p ^{1 - \delta }\), one has
\[\max \left( \left\lvert A+A \right\rvert, \left\lvert A\cdot A \right\rvert \right) \gtrsim \left\lvert A \right\rvert ^{1 + \epsilon (\delta )}\]for some \(\epsilon (\delta ) > 0\).
2. A Helpful Technical Tool
Focusing on the finite field case, the key tool is something called the Plünnecke-Ruzsa calculus, which lets one control the size of rational combinations of sets \(A, B\subseteq \mathbb{F}_p\), assuming that \(A + B\) is comparable in size to \(A\) (more precisely, they are “commensurate”).
The individual inequalities that this tool affords one are not very close at all to the statements above, but they are extremely amenable to being spammed and iterated. Done the right way, this is enough to prove the main theorem. Someone in the audience asked if this was comparable to other “rescale and spam” techniques in analysis, e.g. oscillation decay-like estimates, and I thought this was in general an interesting pattern of argument. Prove something spammable, then spam it.
3. The Power of Dichotomy
Counterexamples to the Bourgain-Katz-Tao theorem would allow one to construct small subfields of \(A\), of which there are only the trivial subfields. The conditions on the cardinality of \(A\) precludes either scenario from happening, and a more sophisticated version of this heuristic known as the “Helfgott pivot argument” powers the proof forward.
There is another variant of the Erdős-Szeméredi conjecture, proven by Bourgain and stated possibly a bit incorrectly:
Theorem 3. Discretised Sum-Product
Let \(A\subseteq \mathbb{R}\) and \(0 < s < 1\). Suppose \(A\) is an “\(s\)-dimensional fractal”. Then for all \(\delta > 0\) there exists \(\epsilon(\delta ) > 0\) such that
\[\max \left( \left\lvert A+A \right\rvert_ \delta , \left\lvert A\cdot A \right\rvert_ \delta \right) \gtrsim \left\lvert A \right\rvert_ \delta ^{1+\epsilon }.\]Here, \(\left\lvert S \right\rvert_ \delta \) is the minimum number of \(\delta \)-intervals needed to cover a set \(S\).
Notably, this theorem is false when one replaces \(\mathbb{R}\) with \(\mathbb{C}\). Taking \(A = \mathbb{R}\) gives an \(s = \frac{1}{2}\)-dimensional subfield, hence the conclusion fails for any \(\delta \).
But naturally I was quite curious about if the trichotomy that all subrings of \(\mathbb{C}\) are half-integer dimensional indicates what a complexified Bourgain’s theorem ought to say:
Question 4.
Does Bourgain’s theorem hold for \(A\subseteq \mathbb{C}\) if we assume \(0 < s < 1\) and \(s \neq \frac{1}{2}\)?