Certified upper bounds for two sum-difference exponents

For finite sets A, B of real numbers and G ⊆ A × B, Katz and Tao proved in 1999 that {a − b : (a, b) ∈ G} has at most N^11/6 elements whenever A, B and {a + b : (a, b) ∈ G} have at most N elements; by an argument of Bourgain, bounds of this kind give lower bounds for the dimension of Kakeya sets. We lower the exponent 11/6 to 1.8273, and to 1.8295 in every abelian group. We also prove |X − Y| ≤ |X + Y|^1.4209 for all finite sets X, Y of integers, improving the exponent 1.4543 of Kleinwaks; through his transfer theorem, this lowers the upper bound for the sum-difference constant of Gyarmati, Hennecart and Ruzsa from 1.3124 to 1.2962. Both proofs add up entropy inequalities among integer combinations of coordinates of several copies of a random pair, with rational weights found by linear programming; all four bounds are also proved in Lean 4.

The certificates, checkers, Lean proofs and programs are archived at doi:10.5281/zenodo.23716681 and available at github.com/tejasvi/sum-difference-exponents (tag v4, commit 9b39a721f255).