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Systems of equations over the group ring of Thompson's group $F$

Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. It is known that if $G$ is amenable then $R$ satisfies the Ore condition: for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It is also true for amenable groups that a non-zero solution exists for any finite system of linear equations over $R$, where the number of unknowns exceeds the number of equations. Recently Bartholdi proved the converse. As a consequence of this theorem, Kielak proved that R.\,Thompson's group $F$ is amenable if and only if it satisfies the Ore condition. The amenability problem for $F$ is a long-standing open question. In this paper we prove that some equations or their systems have non-zero solutions in the group rings of $F$. We improve some results by Donnelly showing that there exist finite sets $Y\subset F$ with the property $|AY| < \frac43|Y|$, where $A=\{x_0,x_1,x_2\}$. This implies some result on the systems of equations. We show that for any element $b$ in the group ring of $F$, the equation $(1-x_0)u=bv$ has a non-zero solution. The corresponding fact for $1-x_1$ instead of $1-x_0$ remains open. We deduce that for any $m\ge1$ the system $(1-x_0)u_0=(1-x_1)u_1=\cdots=(1-x_m)u_m$ has nonzero solutions in the group ring of $F$. We also analyze the equation $(1-x_0)u=(1-x_1)v$ giving a precise explicit description of all its solutions in $K[F]$. This is important since to any group relation between $x_0$, $x_1$ in $F$ one can naturally assign such a solution. So this can help to estimate the number of relations of a given length between generators.