TY - JOUR

T1 - Generalised Rado and Roth criteria

AU - Chapman, Jonathan C

AU - Chow, Sam

PY - 2023/10/1

Y1 - 2023/10/1

N2 - We study the Ramsey properties of equations $a_1P(x_1) + \lcdots + a_sP(x_s) = b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b=a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).

AB - We study the Ramsey properties of equations $a_1P(x_1) + \lcdots + a_sP(x_s) = b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b=a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).

U2 - 10.2422/2036-2145.202302_009

DO - 10.2422/2036-2145.202302_009

M3 - Article (Academic Journal)

SN - 0391-173X

JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

ER -