Let S be the set of numbers that can be expressed as a finite "decimal" fractions in some base with the last non-zero digit taking some fixed value. If you choose the base and the digit appropriately you can easily find sets with the required property. For example the set of finite base 10 decimals ending in 6 is dense round 0 since any real number near zero can be approximated as closely as you like. But if the decimal expansion of x ends in 6 then that of x/2 must end with either 3 or 8 so is not in the set. Another example is base 3 "decimals" ending with 2, in which case x/2 either ends in 1 or is non terminating.