The paper is open access, so the full text is available sans paywall. :) Worth checking out if for no other reason than to appreciate the figures!
https://www.fasebj.org/doi/10.1096/fj.201801412R
It's amazing just how badly this study was distorted by generalized clickbaitism. It chafed me sufficiently enough to put out a video on the subject at the time[1]. The hype really wasn't even necessary... the underlying findings were plenty interesting enough already: that there is an association between stem cell division rate of specific tissues and overall cancer risk. This, however, in no way implies that cancer is only due to bad luck.
Science ended up printing a follow-up actually clarify things, perhaps feeling themselves that the interpretation by the public at large got out of hand a little bit[2].
Interestingly, exercise itself actually helps get an energetically favorable type of energy to the brain in the first place: lactate.[1] Moreover, lactate in the brain triggers the release of BDNF, which causes neurogenesis.[2]
It's really important to not forget the genetic factors at play in Alzheimer's. Between 65-80% of all Alzheimer's patients have at least one genetic version of the ApoE4 allele, a genotype found in around 25% of the population [1]. ApoE is made in astrocytes and is involved in cholesterol transport to neurons and in repairing damage done to neurons that occurs with normal aging. That isn't to say there isn't a dietary link or that critical lifestyle factors don't interact with these genetics, however! Insulin resistance leads to inflammation (which has been shown to accelerate brain aging) and may be one way in which type 2 diabetes is linked with Alzheimer's. I think calling Alzheimer's disease "type 3 diabetes" may be a somewhat dramatic oversimplification and a bit misleading, though, since the mechanism would be a bit more general (in this case).