Sums of four prime cubes in short intervals

Abstract: We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{{3}+p_{2}{3}+p_{3}{3}+p_{4}{3}$,} where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.