A quaternary Diophantine inequality with prime numbers of a special form

Let N be a sufficiently large real number. It is proved here that, for 1 < c < 4803 4040 and for any arbitrary large number E > 0 , the Diophantine inequality | p 1 c + p 2 c + p 3 c + p 4 c - N | < ( log N ) - E is solvable in prime variables p 1 , p 2 , p 3 , p 4 such that, for i = 1 , 2 , 3 , 4 , each of the numbers p i + 2 has at most [ 31540280 12007500 - 10100000 c ] prime factors, counted according to multiplicity. When c → 1 , each p i + 2 is P 16 , which constitutes a large improvement upon the result of Dimitrov [ 14 ] who showed that each p i + 2 is P 32 .