3 papers
math.NT2026
The sum-product conjecture is false for real numbers
Thomas F Bloom, Will Sawin, Carl Schildkraut +1
We disprove the sum-product conjecture for real numbers by constructing arbitrarily large (whose elements are algebraic integers in a number field of degree $…
math.CO2026
Remarks on the disproof of the unit distance conjecture
Noga Alon, Thomas F. Bloom, W. T. Gowers +6
We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the ErdÅs unit distance conjecture, and a sequence of reflections on it. The…
math.NT2026
Odd moments and adding fractions
Thomas F. Bloom, Vivian Kuperberg
We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an applicat…