  We study the problem of estimating the number of points of coincidences of an
idealized gap on the set of integers under a given multiplicative function
$g:\mathbb{N}\longrightarrow \mathbb{C}$ respectively additive function
$f:\mathbb{N}\longrightarrow \mathbb{C}$. We obtain various lower bounds
depending on the length of the period, by varying the worst growth rates of the
ratios of their consecutive values.
