I agree that numbers without a unique representation are dense in R but I believe it is only in so far at the trivial case of the n999... bijection <=> n+1 and

I agree that numbers without a unique representation are dense in R but I believe it is only in so far at the trivial case of the n999... bijection <=> n+1 and the case of numbers which have no representation. I do not think that there are any numbers that have multiple representations outside the n9999... <=> n+1 bijection. There is an argument to be made that 0.9999... is not a valid representation of 1 since it is only convergent to 1 in the limit, where as 1 is 1 all the time.