If my fuzzy math is valid, 31.8mm is about 1.25 inch. I tried on a pill bottle where the diameter measure 1.25 inch and it fits with the smaller rubber shim. I also tried on my seat post which is close to 31.8mm by my caliper measurement and it also fits. You may have to use a layer of old tire tube just to make it a little bit snug as you can see in the picture that the clamping screw is all the way in.

That mount comes with two rubber shims, one is super thick and one is super thin. However the adjusting screw has a lot of give to it as to allow for a wide range of opening.