6 papers · 1 filter
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…
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…
Control and its applications in additive combinatorics
Thomas F. Bloom
We prove new quantitative bounds on the additive structure of sets obeying an 'control' assumption, which arises naturally in several questions within additive combinatorics.…