Has the concept of non-integer $(n+m)$-dimensional spacetime ever been investigated by theoretical physicists? The following image serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime.

The wikipedia article on spacetime, and the sub-article "The priveleged character of $3+1$-dimensional spacetime" in particular, made me think a bit about the possibility that we might live in a non-integer amount of spatial and/or time dimensions. 
The notion of attaching a non-negative real number to a metric space has at least mathematically already been described by such concepts as "Hausdorff Dimension" and "Minkowski-Bouligand Dimension". 
This may sound silly/ignorant/absurd to professional practicing theoretical physicists. To me (a layman), however, it doesn't sound much stranger than the idea of wrapping up six extra dimensions (which is, from what I understand, considered to be a serious possibility by those who study $10$-dimensional String Theory) into intricate shapes called "Calabi-Yau Manifolds". 
Has any research on $(a +b)$-dimensional spacetime (where $a,b \in \mathbb{R}_{\geq 0} $) ever been done? If so, what where the findings? If not, why not? 
 A: yes, in DIMENSIONAL REGULARIZATION dimension is just a parameter and after calculations you set it to $ d=4-\epsilon $ with epsilon tends to 0 so the poles of the Gamma function $ \Gamma (s) $ appear
curiosly enough, if we lived in a world with $ 4.1 $ dimension, then the Gamma function wuold have no poles and the Quantum gravity would be renormalizable.
another question is could the dimension for high energies be only a 'parameter' to be fixed by experiments or by renormalization of the theory ??
A: I believe Gianluca Calcagni does some work in this area. For example, see: http://arxiv.org/abs/1209.1110
And here is a nice blog post on the topic: Dimensional Reduction
A: One example of such an approach is Ambjorn and Loll's Causal Dynamical Triangulations, which is very similar in many ways to the very old idea of Regge calculus, whereby spacetime is discretized.  At small scales, non integer dimensions can emerge.  For an introductory article , see

Jan Ambjørn, Jerzy Jurkiewicz and Renate Loll. The Self-Organizing Quantum Universe. Scientific American (July 2008), 299, pp. 42-49. doi:10.1038/scientificamerican0708-42, available here.

A: A cosmology with multiple time dimensions is only unpredictable if we have the freedom to choose in which direction we move in those time dimensions. But if we had that freedom,  one dimension of time would also be an unpredictable spacetime (i.e.going back in time and killing one's own grandmother, or Michael Faraday or other historically significant person)
If we say however that we are only conscious of mod(t)  ,  i.e. root(t^2 + u^2  + ....)  then there is no causality violation. For example,  if we are smeared over the entire Now circle (in 2t,  or sphere in 3t)  then time angle(s) become irrelevant and only magnitude/  radius of time is what drives physics. That said,  the G.R. solution is not the same as condensing to 1t before solving Einstein's equation for the metric g.
