On character sums over flat numbers

Let q⩾2q⩾2 be an integer, χ be any non-principal character mod q  , and H=H(q)⩽qH=H(q)⩽q. In this paper the authors prove some estimates for character sums of the formW(χ,H;q)=∑n∈F(H)χ(n), whereF(H)={n∈Z|(n,q)=1,1⩽n,n¯⩽q,|n−n¯|⩽H},n¯ is defined by nn¯≡1(modq).