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A function $f$ from $\mathbb{Z}$ to the symmetric matrices over an arbitraryfield $K$ of characteristic $0$ is a $1$-quasihomomorphism if the matrix$f(x+y) - f(x) - f(y)$ has rank at most $1$ for all $x,y \in \mathbb{Z}$. Weshow that any such $1$-quasihomomorphism has distance at most $2$ from anactual group homomorphism. This gives a positive answer to a special case of aproblem posed by Kazhdan and Ziegler.

One-quasihomomorphisms from the integers into symmetric matrices