Let \(f(z) = \sum^\infty_{n = 0} a_n z^{\lambda_n}\), \(\lambda_n \uparrow \infty \), be an entire function and let \(M(r) = \max_\theta |f (re^{i \theta}) |\). Let \(\ln_0 r = r\), \(\ln_{p + 1} r = \ln \ln_p r\). The numbers \[ \rho = \rho_f = \varlimsup_{r \to \infty} {\ln_p M(r) \over \ln_qr}, \quad T_f= \varlimsup_{r \to \infty} {\ln_{p - 1} M(r) \over (\ln_{q - 1} r)^\rho}, \quad t_f= \varliminf_{r\to \infty} {{\ln_{p-1} M(r)} \over {(\ln_{q-1} r)^\rho}} \] are called the \((p,q)\) order, type and lower type of \(f\). In the joined papers (1975, 1977) of S. K. Bajpai, G. P. Kapoor, O. P. Juneja it was proved that if \(0 \leq t_f < \beta < T_f < \infty\) then \(f = g + h\), where \(T_g \leq \beta\) and the function \(h(z) = \sum^\infty_{k = 0} b_k z^{\lambda_{n_k}}\) \((b_k \neq 0)\) has a special property. In the case \(p = 2\), \(q \neq 2\) this property is the following \(t_h \geq \beta \varliminf_{r \to \infty} {\lambda_{n_k} \over \lambda_{n_{k + 1}}}\). The authors prove the spread variant of the theorem applying a proximate \((p,q)\) order \(\rho (r)\). Further, the authors study relations between asymptotic properties of \(M(r)\) and properties of the sequences \(a_n\), \({a_{n - 1} \over a_n}\).