Frame dimension functions and phase retrievability

The frame dimension function of a frame \({{\mathcal {F}}}= \{f_j\}_{j=1}^{n}\) for an n-dimensional Hilbert space H is the function \(d_{{{\mathcal {F}}}}(x) = \dim {\textrm{span}}\{ \langle x, f_{j}\rangle f_{j}: j=1,\ldots , N\}, 0\ne x\in H.\) It is known that \({{\mathcal {F}}}\) does phase retrieval for an n-dimensional real Hilbert space H if and only if \({\textrm{range}} (d_{{{\mathcal {F}}}}) = \{ n\}.\) This indicates that the range of the dimension function is one of the good candidates to measure the phase retrievability for an arbitrary frame. In this paper we investigate some structural properties for the range of the dimension function, and examine the connections among different exactness of a frame with respect to its PR-redundance, dimension function and range of the dimension function. A subset \(\Omega \) of \(\{1,\ldots , n\}\) containing n is attainable if \({\textrm{range}} (d_{{{\mathcal {F}}}}) = \Omega \) for some frame \({{\mathcal {F}}}.\) With the help of linearly connected frames, we show that, while not every \(\Omega \) is attainable, every (integer) interval containing n is always attainable by an n-linearly independent frame. Consequently, \({\textrm{range}}(d_{{{\mathcal {F}}}})\) is an interval for every generic frame for \({\mathbb {R}}\,^n.\) Additionally, we also discuss and post some questions related to the connections among ranges of the dimension functions, linearly connected frames and maximal phase retrievable subspaces.