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Sum-product, unit distances, and number fields
By Thomas Bloom<br>on 31 May 2026
In this blog post I will give my personal view on the recent counterexamples to the unit distance conjecture and sum-product conjecture over the reals (see [90] and [52] respectively). My goal is to sketch the constructions and try and give some intuition as to where they came from and why they work. My main target audience is the me-of-a-month-ago, who did not know much algebraic number theory, and who needs the relevant parts of the basic theory in this area explained, but wants to know exactly where the quantitative improvements come from. (I know a bit more algebraic number theory now, but still much less than I'd like!) My focus is on the combinatorial side, and I will stop with an appeal to the literature as soon as we need to do any serious number theory. (In particular I will not attempt to discuss even the statement, let alone the proof, of the Golod-Shafarevich theorem.)
Any faults in this post are entirely my own, and if you are confused by any aspect of the proofs sketched here I encourage you to review the original sources. Comments and corrections are welcome in the comment section.
The original OpenAI disproof of the unit distance conjecture can be read here, with a human-written companion paper here and an explicit (and improved) version of the argument by Sawin here. The disproof of the sum-product paper by me, Sawin, Schildkraut, and Zhelezov, is here.
Warmup round
Consider the following natural statement in additive combinatorics: if $A+A=\{ a+b : a,b\in A\}$ and $A-A=\{a-b : a,b\in A\}$ then how well can we bound $\lvert A+A\rvert$ above by $\lvert A-A\rvert$? In other words, for what constant $c$ can we say $\lvert A+A\rvert \leq \lvert A-A\rvert^c$? This is trivially true with $c=2$, since $\lvert A-A\rvert\geq \lvert A\rvert$ and $\lvert A+A\rvert\leq \lvert A\rvert^2$. Also obviously, weird things might happen for some small sets. But is it possible with $c$ arbitrarily close to $1$ - in other words, with $c=1+\epsilon$, provided $\lvert A\rvert$ is large enough in terms of $\epsilon$? Initial experimentation suggests that it might be - the sum set tends to be smaller than the difference set, and in all the natural examples one might consider, anything that forces $A+A$ to be large also forces $A-A$ to be large.
So maybe one could conjecture that $\lvert A+A\rvert \leq \lvert A-A\rvert^{1+o(1)}$. This is false, however, thanks to the 'tensor power trick'. First note that I never actually said where the $A$ lives - maybe you were thinking of a set of integers, but in additive combinatorics all of these concepts make sense in any abelian group, so we can also consider finite $A\subset \mathbb{Z}^d$ for arbitrarily large $d$.
How does this help? Well first find some example where $A+A$ is large compared to $A-A$, just by fluke/law of small numbers. For example, if $A=\{0, 2, 3, 4, 7, 11, 12, 14\}$ then $A+A$ has $26$ elements but $A-A$ has $25$ elements. So here $\lvert A+A\rvert \geq \lvert A-A\rvert^c$ where $c=\log_{25}(26)\approx 1.012$. So what, you might say - this is one particular finite $A$, and I knew that small sets might be weird, so I covered this by saying 'exponent at most $1+\epsilon$ only for sufficiently large sets $A$. The point is that we can blow up any fixed example like this to arbitrary large examples with the same behaviour: if we let $B=A^d\subset \mathbb{Z}^d$ then the sum and difference sets are also just the cartesian products, so $\lvert B+B\rvert=26^d$ and $\lvert B-B\rvert =25^d$. This means that there exist arbitrarily large sets $B$ (in some abelian group) such that\[\lvert B+B\rvert \geq \lvert B-B\rvert^{1.012\cdots}.\]Thus the initial naive guess was wrong; the best exponent is $>1$ by at least $0.012$, and no restricting to 'sufficiently large sets' is going to save it.
(If you're interested in this type of problem, and earlier examples of this kind of tensor power construction used in additive combinatorics, see e.g. this page or [GRH07] by Gyarmati, Ruzsa, and Hennecart.)
The constructions in both the sum-product and unit distance counterexamples have a similar flavour: one finds a 'trivial' construction, only winning by some constant factor, and then 'blowing up' this constant win by taking $d$-dimensional versions, where $d\to \infty$. In the example above this was easy to do, since given any abelian group $G$ we can just form the direct product $G^d$ and everything scales as expected. In the sum-product and unit distance problems, however, we have to (a) construct sets in $\mathbb{R}$ and $\mathbb{R}^2$ respectively, not in some $\mathbb{R}^d$, and (b) make sure that not just addition...