On some arithmetic conditions of recurrent sequences modulo prime p

We study the $K$-Fibonacci sequence ${\mathcal{F}}_p$ modulo prime $p$. Cardinalities of sets $\mathrm{\mid }{\mathcal{F}}_p+{\mathcal{F}}_p\mathrm{\mid }$ and $\mathrm{\mid }{\mathcal{F}}_p\cdot {\mathcal{F}}_p\mathrm{\mid }$ are estimated. We present the method of estimating doubling constant of some $m$-dimensional recurrent sets in ${\mathbb{F}}_p$.