One useful characterization is: if and only if the coefficients in the continued fraction expansion of $\xi$ are bounded ($c_1$ and $c_2$ can be bounded in terms of the minimal and maximal coefficient). In particular, this is true when the expansion is periodic, i.e. when $\xi$ is a quadratic irrational.

One more useful pieceful information: if $c_1>1/3$, then there are countably many such $\xi$'s (each equivalent to a quadratic irrational occurring in the Markoff spectrum), while if $c_1=1/3$, then there are already continuum many (pairwise inequivalent) such $\xi$'s. Here equivalence is meant under the action of $\mathrm{SL}_2(\mathbb{Z})$ by fractional linear transformations.

One useful characterization is: if and only if the coefficients in the continued fraction expansion of $\xi$ are bounded ($c_1$ and $c_2$ can be bounded in terms of the minimal and maximal coefficient). In particular, this is true when the expansion is periodic, i.e. when $\xi$ is a quadratic irrational.

One more useful pieceful information: if $c_1>1/3$, then there are countably many such $\xi$'s (each equivalent to a quadratic irrational occurring in the Markoff spectrum), while if $c_1=1/3$, then there are already continuum many (pairwise inequivalent) such $\xi$'s. Here equivalence is meant under the action of $\mathrm{SL}_2(\mathbb{Z})$ by fractional linear transformations.

Clarification. My response focused on a classical version of the problem, in which we try to achieve the lower bound for all $q$'s with $c_1$ as large as possible, while we try to achieve the upper bound for infinitely many $q$'s with $c_2$ as small as possible.

One useful characterization is: if and only if the coefficients in the continued fraction expansion of $\xi$ are bounded ($c_1$ and $c_2$ can be bounded in terms of the minimal and maximal coefficient). In particular, this is true when the expansion is periodic, i.e. when $\xi$ is a quadratic irrational.

One more useful pieceful information: if $c_1>1/3$, then there are countably many such $\xi$'s (each equivalent to a quadratic irrational occurring in the Markoff spectrum), while if $c_1=1/3$, then there are already continuum many such $\xi$'s.

One useful characterization is: if and only if the coefficients in the continued fraction expansion of $\xi$ are bounded ($c_1$ and $c_2$ can be bounded in terms of the minimal and maximal coefficient). In particular, this is true when the expansion is periodic, i.e. when $\xi$ is a quadratic irrational.

One more useful pieceful information: if $c_1>1/3$, then there are countably many such $\xi$'s (each equivalent to a quadratic irrational occurring in the Markoff spectrum), while if $c_1=1/3$, then there are already continuum many (pairwise inequivalent) such $\xi$'s. Here equivalence is meant under the action of $\mathrm{SL}_2(\mathbb{Z})$ by fractional linear transformations.