On Rings of Numbers Which Are Normal To One Base But Non-normal to Another

AbstractFor any natural number g ≥ 2, and for any odd prime p not dividing g, we give an explicit example of a ring R of real numbers with the Following three properties: •is uncountable,•all numbers ∈ , ≠ 0, are normal to base , and•all numbers ∈ are non-normal to base · . This result contains, e.g., explicitly given numbers x such that, for any non-constant polynomial q ∈ Z[x], the number q(x) is normal to base 3 but non-normal to base 15