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Blog - Sum-product, unit distances, and number fields | Erdős Problems

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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

Blog - Sum-product, unit distances, and number fields | Erdős Problems Forum Inbox Favourites Tags More FAQ Prizes Problem Lists Definitions Links Forum Menu Inbox Favourites Tags FAQ Prizes Problem Lists Definitions Links Go Go Dual View Random Solved Random Open Sum-product, unit distances, and number fields By Thomas Bloom 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 a

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