On cardinality of character sums with Beatty sequences associated with composite modules

Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of character sums and their estimation with respect to non homogeneous Beatty sequences $\beta_\alpha = \lfloor \alpha n+ \beta :n = 1, 2, 3 ...\rfloor$ where $ \beta $ in real numbers and $\alpha $ greater than zero is irrational. In order to estimate the cardinality, the discrepancy is used to measure the number of uniform distribution for Beatty sequences. Pigeonhole principle is discussed on the estimation of the fractional part of Beatty sequences involve. Meanwhile, Cauchy inequalities is applied to expand the double character sums. Then, the cardinality of double character sums is obtained by applying the extension properties of additive and multiplicative character sums. The result obtained is depend on the existing of identity of additive and multiplicative character sums and the uniformly distribution modulo $1$. The result of the estimation in this study over composite modules is more general compared to previous studies, which only cover prime modules.