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Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices,labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposableregular modules with preprojective Gabriel-Roiter submodules, in particular,those $\tau^{-i}X$ with $\udim X=(1,c)$ for $i\geq 0$ and some $1\leq c\leqn-1$ will be studied. It will be shown that for each $i\geq 0$ the irreduciblemonomorphisms starting with $\tau^{-i}X$ give rise to a sequence ofGabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of thoseproduce a sequence of direct successors. In particular, there are infinitelymany GR-segments, i.e., a sequence of Gabriel-Roiter measures closed underdirect successors and predecessors. The case $n=3$ will be studied in detailwith the help of Fibonacci numbers. It will be proved that for a regularcomponent containing some indecomposable module with dimension vector $(1,1)$or $(1,2)$, the Gabriel-Roiter measures of the indecomposable modules areuniquely determined by their dimension vectors.

Regular modules with preprojective Gabriel-Roiter submodules over $n$-Kronecker quivers