4 research outputs found
Equation-regular sets and the Fox–Kleitman conjecture
Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero
integer b such that the 2k-variable linear Diophantine equation
∑k
i=1
(xi − yi) = b
is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all
b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for
all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of
the corresponding equation for all b ≥ 1. In particular, this independently confirms the
conjecture for k = 3. We also briefly discuss the case k = 4
On the degree of regularity of a certain quadratic Diophantine equation
We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic
Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic
progressions