The order of inverses mod q

Let q be a prime number and let a = ( a 1 , …, a s ) be an s ‐tuple of distinct integers modulo q . For any x coprime with q , let 1 ⩽ x ¯ < q be such thatx ¯ x = 1   ( mod   q ) . For fixed s and q →∞ an asymptotic formula is given for the number of residue classes x modulo q for whichx + a 1 ¯ < x + a 2 ¯ < ⋯ < x + a s . ¯The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q ], is also treated.