In this paper, we consider concerning absolute convergence
of the lacunary Fourier series 
$$
\displaystyle{
f(x)\sim \sum_{k=1}^{\infty} 
\left( a_{n_k} \cos n_k x + b_{n_k} \sin n_k x \right)
\hspace{1mm},}
$$
where $\left\{ n_k \right\} ( k \in$~{\bf N}) \;  
is a strictly increasing sequence of natural numbers. 
Under some condition regarding a sequence 
$\left\{ n_k \right\}$, 
the lacunary Fourier series converges absolutely.
We generalize known results by using 
a new modulus of continuity.