Boundedness of solutions of the first-order linear multidimensional difference equations

We investigate the boundedness of solutions of the first order linear difference equation of the form $x_{n+1} = Ax_{n} + y_{n}, \; n \geq 1$ where $A$ is a square matrix with complex entries, sequence $\{y_{n}\}_{n\geq 1}$ and initial value $x_1$ are supposed to be known. Firstly, we discuss the one-dimensional case of this equation $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ where $a$ is a complex number. In particular, we obtain the sufficient conditions for boundedness or unboundedness of the solutions in case $|a|=1$(the critical case) by considering the exponential sums of the forms $\sum y_{n}e(n\varphi)$ and $\sum e(f(n))$. Then we proceed to the investigation of the equation in the multidimensional case and reduce our problem to analysis of the spectrum and Jordan cells of matrix $A$. The problem is especially interesting when spectrum of $A$ contains eigenvalues $\lambda$ with $|\lambda|=1$. At the end of the article we obtain a theorem that reveals the connection between equations $x_{n+1} = ax_{n} + y_{n}, \; n \geq 1$ with $|a|=1$ and $x_{n+1} = Jx_{n} + y_{n}, \; n \geq 1$ with $J$ being a Jordan cell of an eigenvalue $\lambda$, $|\lambda|=1$.