On rational approximations to an irrational number

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Let α be a real irrational number,J and subinterval of (0, 1), and |J| the lenght ofJ. For every integern, let us denote byR(να) the residues mod. 1 of the real numbers ν α (ν=1,2...,n) and byN(n, J) the number of theR(ν α)’s lying inJ. Set $$A*\left( x \right): = Sup \left| n \right|\left. J \right| - \left. {N\left( {n,J} \right)} \right|$$ the supremum being taken over alln between 1 andx and all the subintervalsJ of (0, 1). The main result in this paper is the functional inequality $$A*\left( x \right) \leqq A*\left( \sigma \right) + 2xU\left( \sigma \right) \left( {x > 0, \sigma > \sigma _0 } \right)$$ whereU (∂) somehow depends on the function $$T\left( \sigma \right): = Min\left| {z_1 \alpha + z_0 } \right|, \left| {z_1 } \right| \leqq \sigma , z_0 \wedge z_1 \in \mathbb{Z}$$ . As a consequence both new and well known results can be obtained on the order of magnitude ofA*(x)