A study of the modifications of the magnetic properties of Ho$_{2-x}$Y$_x$Sn$_2$O$_7$ upon varying the concentration of diamagnetic Y$^{3+}$ ions is presented. Magnetization and specific heat measurements show that the Spin Ice ground-state is only weakly affected by doping for $xleq 0.3$, even if non-negligible changes in the crystal field at Ho$^{3+}$ occur. In this low doping range $mu$SR relaxation measurements evidence a modification in the low-temperature dynamics with respect to the one observed in the pure Spin Ice. For $xto 2$, or at high temperature, the dynamics involve fluctuations among Ho$^{3+}$ crystal field levels which give rise to a characteristic peak in $^{119}$Sn nuclear spin-lattice relaxation rate. In this doping limit also the changes in Ho$^{3+}$ magnetic moment suggest a variation of the crystal field parameters.
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