7 papers
Remarks on the inverse Littlewood conjecture
Thomas F. Bloom, Ben Green
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if is a finite set of integers with then $\…
More on the sum-product problem for integers with few prime factors
Rishika Agrawal, Thomas F. Bloom, Giorgis Petridis
We show that if is a finite set of integers in which every integer is divisible by many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lver…
A polynomial Freiman-Ruzsa inverse theorem for function fields
Thomas F. Bloom
Using the recent proof of the polynomial Freiman-Ruzsa conjecture over by Gowers, Green, Manners, and Tao, we prove a version of the polynomial Freiman-Ruzsa conje…
Integers with small digits in multiple bases
Thomas F. Bloom, Ernie Croot
We show that, for any , if are distinct coprime integers, sufficiently large depending only on , then for any there are infinitely many integers…
Additive structure in convex sets
Thomas F. Bloom, Jakob Führer, Oliver Roche-Newton
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set containing …
The Bombieri-Pila determinant method
Thomas F. Bloom, Jared Duker Lichtman
We give a concise and accessible introduction to the real-analytic determinant method for counting integral points on algebraic curves, based on the classic 1989 paper of Bombieri…