I'm not recommending a ketogenic diet. I was on one for many years. As with the study results, it helped, but my symptoms still progressed. I believe this is because it restricts gluten but does not eliminate it.

A gluten free diet eliminates wheat, barley and rye products, not carbohydrates in general.
Fruits and veggies are totally legal. As are potatoes, corn, and rice. As a matter of fact, sugar is too!

I'm sorry if that wasn't clear.