a finite group $G$ of order $m$ relative to a (forbidden) subgroup $U$ is a $k$-subset $R$ with the property that $$\delta(g)=\begin{cases} 
    k & g=1_G \\
    \lambda & g\in G \backslash U\\
    0 & otherwise \\
    \end{cases}$$ The RDS is called [*semiregular*]{} if $m = k = u \lambda$.

Clearly a $(v,k,\lambda)$-DS is a $(v,k; k\lambda, n+k\lambda)$-PGDS and an $(m,u,k,\lambda)$ semiregular RDS is a $(mu,k; \lambda(k-1), k(\lambda+1)-\lambda)$-PGDS [@olmez1].

Group characters are powerful objects to investigate various types of difference sets. A [*character*]{} $\chi$ of an abelian group $G$ is a homomorphism from $G$ to the multiplicative group of the complex numbers. The character $\chi_0$ defined by $\chi_0(g) = 1$ for all $g \in G$ is called the [*principal character*]{}; all other characters are called [*nonprincipal*]{}. We define the character sum of a subset $S$ of an abelian group $G$ as $\chi(S):= \sum_{s \in S} \chi(s)$.

\[chartheoryPGDS\] A $k$-subset $S$ of an abelian group $G$ is a partial geometric difference set in $G$ with parameters $(v,k;\alpha,\beta)$ if and only if $|\chi (S)|=\sqrt{\beta-\alpha}$ or $\chi (S)=0$ for every non-principal character $\chi$ of $G$.

For instance, let $f$ be a $p$-ary bent function from the field $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$. The set $G_f=\{(x,f(x)): x\in 