"Any arbitrary" versus "An arbitrary"

There is one context I can think of where the two sentences will differ significantly: if the action (planting bushes?) is repeated.

An arbitrary distance implies that you chose one arbitrary distance for spacing all of them.

while

Any arbitrary distance tends to allow for different distances being selected for each spacing.

"Implies" and "tends to" are my weasel words for admitting that these are not hard-and-fast differences. Either interpretation is possible, but I would tend to use the an/any distinction myself to indicate the (lack of) variation.

I prefer #2, in my opinion it flows better. The stress falls on roots and arbitrary which together carry most of the meaning of your sentence.

I noted also that more in 'more redundant' is itself redundant.

Just thinking about redundancy in speech and writing, it's something I try hard to avoid when writing technical documents. I prefer to find the most accurate and concise expressions I can. In everyday writing, however, it is often tempting to use redundancy for rhetorical purposes or preserve rhythm.

I'm interested in the context of this sentence. It seems that "arbitrary" is entirely redundant, or at least seems incompatible with "could". I am guessing that your intent is either:

"The roots could be any distance apart."

"The roots are an arbitrary distance apart."

When I read "the roots could be an arbitrary distance apart", I get the implication, "the roots are a certain distance apart, and the distance might be determined by a logical rule, or the distance might be determined entirely capriciously."

"Could" here seems to refer the fact that you are not sure if the distance is arbitrary, rather than the fact that you are not sure what the distance is.

But maybe that is your intent, or maybe I am off base.

I'm with z7sg in thinking the second form ("an") flows better. That's not the whole story, though; the "any" version does add considerable emphasis to the arbitrariness of choice of roots, so may be more appropriate if that is the point you want to bring out.