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In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain 
$\unicode[STIX]{x1D709}$ depending on the dimension of the space, 
$\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$ almost everywhere and 
$\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$ for almost all 
$x$ and 
$y$. This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.
