Three properties of dynamical systems (recurrence, connectivity and proximality) are quantified by introducing and studying the gauges (measurable functions) corresponding to each of these properties. The properties of the proximality gauge are related to the results in the active field of shrinking targets.The emphasis in the present paper is on the IETs (interval exchange transformations) \(( \mathcal {I},T)\), \(\mathcal {I}=[0,1)\). In particular, we prove that if an IET T is ergodic (relative to the Lebesgue measure λ), then the equality $$ \liminf_{n\to\infty} \, n\, \bigl|T^n(x)-y \bigr|=0 $$ (A1) holds for λ×λ-a.a. \((x,y)\in \mathcal {I}^{2}\). The ergodicity assumption is essential: the result does not extend to all minimal IETs. Also, the factor n in (A1) is optimal (e.g., it cannot be replaced by n ln(ln(lnn))).On the other hand, for Lebesgue almost all 3-IETs \(( \mathcal {I},T)\) we prove that for all ϵ>0 $$ \liminf_{n\to\infty} \, n^ \epsilon \bigl |T^n(x)-T^n(y)\bigr| = \infty,\quad\text{for Lebesgue a.a.} \ (x,y)\in \mathcal {I}^2. $$ (A2) This should be contrasted with the equality lim inf n→∞ |T n(x)−T n(y)|=0, for a.a. \((x,y)\in \mathcal {I}^{2}\), which holds since \(( \mathcal {I}^{2}, T\times T)\) is ergodic (because generic 3-IETs \(( \mathcal {I},T)\) are weakly mixing).We introduce the notion of τ-entropy of an IET which is related to obtaining estimates of type (A2).We also prove that no 3-IET is strongly topologically mixing.