Sum-product, unit distances, and number fields
4 days ago
- #algebraic number theory
- #unit distance conjecture
- #combinatorial counterexamples
- The author provides a personal overview of recent counterexamples to the unit distance and sum-product conjectures over the reals, focusing on constructions and intuition.
- The target audience is the author's past self, requiring basic algebraic number theory explanations while emphasizing quantitative improvements and combinatorial aspects.
- The unit distance conjecture disproof by OpenAI, with companion papers and improvements, and the sum-product disproof by the author and colleagues are referenced.
- A warmup example in additive combinatorics demonstrates the tensor power trick to show that |A+A| can be superlinearly bounded by |A-A| using Cartesian products in higher dimensions.
- Both counterexamples use a similar strategy: find a trivial construction with a constant improvement, then amplify it via high-dimensional analogs using number fields.
- An algebraic number theory refresher covers number fields, embeddings, the ring of integers as a lattice, the unit group, and key parameters like discriminant and regulator.
- For the sum-product counterexample, sets are constructed using additive and multiplicative balls in totally real number fields, combining arithmetic and geometric progressions.
- The construction yields sets A with both sum and product sets bounded by |A|^{2-c}, requiring number fields with bounded discriminant, provided by Golod-Shafarevich theorem via Martinet.
- For the unit distance counterexample, a high-dimensional analog uses number fields with a quadratic extension K(i) and units of norm 1 to generate many unit distance pairs.
- The construction produces point sets P in ℝ² with n points and at least n^{1+c} unit distances, again relying on towers of number fields with controlled parameters.
- The key novelty is applying existing number field towers from Golod-Shafarevich to combinatorial problems, using shallow algebraic number theory for the reductions.
- The AI's discovery process involved persistent exploration and literature synthesis, contrasting with human research tendencies, but many AI attempts likely fail silently.