Rewinder for the production of logs of paper material, comprising a winding station for winding the paper with a first winding roller and a second winding roller adapted to delimit, with their respective external surfaces, a nip through which a paper web consisting of one or more paper plies is fed and is intended to be wound in said station to form a log, and a third winding roller that, in relation to a feeding direction of the web, is positioned downstream of the first and the second winding rollers (R1, R2), wherein the second winding roller is positioned at a lower level than the first winding roller, wherein said first, second and third winding rollers rotate around respective axes of rotation, wherein the axes of rotation of the first winding roller, the second winding roller and the third winding roller are horizontal and parallel to each other and are oriented transversely to said feeding direction, wherein the third winding roller is connected to an actuator which allows it to be moved cyclically from and to said nip so that the position of the winding third roller varies in relation to the other two winding rollers (R1, R2) during the production of the logs, and wherein each of said winding rollers (R1, R2, R3) rotates around its own axis being connected to a corresponding electric motor (M1, M2, M3), characterized in that: - the rewinder comprises a detection system with optical means (5, 50) adapted to detect, in a succession of predetermined detection times, a succession of diameters assumed in such times by a log being formed in the winding station, and a programmable electronic unit that is connected to said electric motors (M1, M2, M3) and to said optical means (5, 50), - the programmable electronic unit is programmed to compare said succession of diameters measured by the optical means (5, 50) with a corresponding succession of preset diameters and to calculate a sequence of differences (e1, e2,..., en) between said measured and preset diameters (DE, DT); - said programmable electronic unit receives from said optical means a signal relating to the value of the actual diameter of the completed log i.e. of the diameter assumed by each log when the winding of the latter is completed; - said programmable electronic unit determines a parameter (a) related to the trend over time of the values (e1, e2,..., en) forming said sequence of differences; - said programmable electronic unit changes the relative speed of said first and second roll (R1, R2) depending on the value of said parameter (a); - said processing unit is connected to means for adjusting the thickness (T3; EM; EX) of the paper web reaching the winding station so as to effect an automatic adjustment of said thickness acting on said adjustment means (T3; EM; EX) if the parameter (a) is between predetermined values 2. aN and aP, where aN and aP are the ends of a range of values containing the zero value; - said automatic adjustment is performed on the basis of the diameter of the completed log as detected by said optical means (5, 50) by decreasing or increasing the thickness of the paper web on the basis of detected diameter of the completed log when the detected diameter of the completed log is, respectively, higher or lower than a preset diameter value. - the rewinder comprises a detection system with optical means (5, 50) adapted to detect, in a succession of predetermined detection times, a succession of diameters assumed in such times by a log being formed in the winding station, and a programmable electronic unit that is connected to said electric motors (M1, M2, M3) and to said optical means (5, 50), - the programmable electronic unit is programmed to compare said succession of diameters measured by the optical means (5, 50) with a corresponding succession of preset diameters and to calculate a sequence of differences (e1, e2,..., en) between said measured and preset diameters (DE, DT); - said programmable electronic unit receives from said optical means a signal relating to the value of the actual diameter of the completed log i.e. of the diameter assumed by each log when the winding of the latter is completed; - said programmable electronic unit determines a parameter (a) related to the trend over time of the values (e1, e2,..., en) forming said sequence of differences; - said programmable electronic unit changes the relative speed of said first and second roll (R1, R2) depending on the value of said parameter (a); - said processing unit is connected to means for adjusting the thickness (T3; EM; EX) of the paper web reaching the winding station so as to effect an automatic adjustment of said thickness acting on said adjustment means (T3; EM; EX) if the parameter (a) is between predetermined values 2. aN and aP, where aN and aP are the ends of a range of values containing the zero value; - said automatic adjustment is performed on the basis of the diameter of the completed log as detected by said optical means (5, 50) by decreasing or increasing the thickness of the paper web on the basis of detected diameter of the completed log when the detected diameter of the completed log is, respectively, higher or lower than a preset diameter value.