This paper generalizes some results from another one, namely [3]. We have studied the issues of expressing natural numbers as a sum of powers of natural numbers in paper [3]. It means we have studied sets of type $A=\{n_1^{k_1}+n_2^{k_2}+\cdots +n_m^{k_m}\mid n_i\in \mathbb{N}\cup \left\{0\right\},i=1,2\cdots ,m,(n_1,n_2,\cdots ,n_m)\ne (0,0,\cdots ,0)\},$ where $k_1,k_2,\cdots ,k_m\in \mathbb{N}$ were given natural numbers. Now we are going to study a more general case, i.e. sets of natural numbers that are expressed as sum of integral parts of functional values of some special functions. It means that we are interested in sets of natural numbers in the form $$k=\left[f_1\left(n_1\right)\right]+\left[f_2\left(n_2\right)\right]+\cdots +\left[f_m\left(n_m\right)\right].$$

Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that $\left|\Sigma \left(X\right)\right|\ge 2^{r-1}\left(\right|X|-r+2)-1$ for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.