We establish upper bounds for the number of primitive integer solutions to inequalities of the shape 0 < |F (x, y)| ≤ h, where F (x, y) = (αx+βy) r −(γx+δy) r ∈ Z[x, y], α, β, γ and δ are algebraic constants with αδ −βγ = 0, and r ≥ 5 and h are integers. As an important application, we pay special attention to binomial Thue's inequaities |ax r − by r | ≤ c. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.

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We give improved upper bounds for the number of solutions of the Thue equation F(x, y) = h where F is an irreducible binary form of degree ≥ 3.

Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.For\alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2,let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers with\gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1We prove that there exists δ> 0 such that for all integers r1,r2,\matrix{ {\left| {\left\{ {0 \le n < N:{S_q}(n) \equiv {r_1}\left( {\bmod \,{m_1}} \right),\,\,{S_\alpha }(n) \equiv {r_2}\left( {\bmod \,{m_2}} \right)} \right\}} \right|} \cr { = {N \over {{m_1}{m_2}}} + 0\left( {{N^{1 - \delta }}} \right).} \cr } The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case \alpha = \left[ {\bar 1} \right] by Spiegelhofer.