29 citations · 64 across the 16 of their papers we have counts for
5 papers · 1 filter
Stronger sum-product inequalities for small sets
Misha Rudnev, George Shakan, Ilya Shkredov
Let be a field and a finite be sufficiently small in terms of the characteristic of if . We strengthen the "threshold" sum-product inequality $$|AA|^3…
Breaking the 6/5 threshold for sums and products modulo a prime
G. Shakan, I. D. Shkredov
Let of size at most . We show for . Our main tools are the cartesian product point--line incide…
Monochromatic Hilbert cubes and arithmetic progressions
József Balogh, Mikhail Lavrov, George Shakan +1
The Van der Waerden number denotes the smallest such that whenever is --colored there exists a monochromatic arithmetic progression of length . Similarly,…
On higher energy decompositions and the sum-product phenomenon
George Shakan
Let be finite. We quantitatively improve the Balog-Wooley decomposition, that is can be partitioned into sets and such that $$\max\{E^+(B) , E^{\…
Fractal solutions of dispersive partial differential equations on the torus
Burak Erdoğan, George Shakan
We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Sc…