Uniform distribution of sequence mod 1

Question

Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?

Answer 1

Yes, this is true. Because of the ergodicity of the doubling map on $\mathbb{R}/\mathbb{Z}$, for any interval $I$ of $[0,1]$ and for $\lambda$-almost every $x\in [0,1]$, we have $\displaystyle \frac{1}{N} \sum_{k=1}^N \mathbb{1}_I(2^k x \text{ mod } 1) \to \lambda(I)$, where $\lambda$ denotes for the Lebesgue measure on $[0,1]$. Let $(I_n)_{n=1}^{\infty}$ be the countable collection of subintervals of $[0,1]$ with rational endpoints, and $(E_n)_{n=1}^\infty$ be the corresponding exceptional subsets of $[0,1]$ where the above convergence does not happen. Let $E=\cup_{n=1}^\infty E_n$. Then for any $x\in [0,1]\setminus E$ and for an arbitrary interval $J\subset [0,1]$, we have $\displaystyle \frac{1}{N} \sum_{k=1}^N \mathbb{1}_J(2^k x \text{ mod } 1) \to \lambda(J)$. Since each $E_n$ has measure $0$, so does $E$, and this completes the proof of the statement.

Edit: As noted by Anthony Quas, the earlier argument was incomplete.

Answer 2

It is unclear what is "most $r$'s even mean. A standard argument would show that for any increasing sequence, for Lebesgue almost every $x$, $a_{n}.x$ is equidistributed mod $1$.

For the case of powers of $2$, as mentioned, it is an easy consequence of the ergodicity of the $\times 2$ action with respect to the Lebesgue measure.

The state of the art is a theorem by Host (with subsequent proofs by Hochman and others) showing that if $r$ is sampled at random for a $\times 3$ (say) ergodic measure with positive entropy (for which the Lebesgue measure is one such an example) then its orbit under $\times 2$ is equidistributed with respect to the Lebesgue measure. This theorem shows for example that generic numbers from the middle thirds Cantor set are equdistributed under the $\times 2$ action...

Answer 3

In other words the question is: are almost all numbers normal?

The answer is "yes" and this is the beginning of the theory of normal numbers. One can easely find the early history of normal numbers in Wikipedia:

The concept of a normal number was introduced by Émile Borel (1909). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal, establishing the existence of normal numbers.