On measure-preserving $$\mathbb{F}_{p}^{\omega}$$-systems of order k

Abstract Building on previous work in the nilspace-theoretic approach to the study of Host–Kra factors of measure-preserving systems, we prove that every ergodic $$\mathbb{F}_{p}^{\omega}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> -system of order k is a factor of an Abramov $$\mathbb{F}_{p}^{\omega}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> -system of order k . This answers a question of Jamneshan, Shalom and Tao.