Very thinly slice 1 lb. of potatoes. Make a layer of potatoes on top of the leeks. Sprinkle with salt and pepper.
Slice 1/2 lb. turnips thinly, but not as thinly as the potatoes. Make a layer atop the potatoes. Sprinkle with salt and pepper and drizzle with 1 t. olive oil.
Grate 8 oz. gruyere cheese. Make a thin layer atop the turnips.
Layer the remaining vegetables, sprinkling with salt and pepper and drizzling with a little more olive oil.
Top with the remainder of the cheese.
Bake the gratin at 350 degrees until the top and edges are nicely browned.