How big can $\{t \in [0,1] : \lim_{n\to\infty}\exp(2\pi itk_n)=1\}$ be? Let $(k_n)_{n\geq1}$ be a incresing sequence of positive integers. How big can be
$$C = \{t \in [0,1] : \lim_{n\to\infty}\exp(2\pi itk_n)=1\}?$$
My thoughts so far:


*

*If $k_n = n!$, then $C \supseteq \mathbb{Q}$, so $C$ can be a dense in $[0,1]$.

*More generally, if $Q\subseteq [0,1]$ is any countable set, then $C\supseteq Q$ for some choice of $(k_n)_{n\geq1}$ .

*By compactness, one can take a subnet $(k_\lambda)_{\lambda\in\Lambda}$ of $(k_n)_{n\geq1}$ such that $\lim_{\lambda\to\infty}\exp(2\pi itk_\lambda)$ exists for all $t \in [0,1]$. I don't know if it is possible to make this limit equal to $1$. In any case, I am not allowing nets in the definition of $C$.

*If $k_n = n$, then $C = \{1\}$.


To make the question more concrete: can $C$ be uncountable? If the answer is yes, can it have positive Lebesgue measure?
 A: It is indeed possible to have $C$ be uncountable.  Since $|e^{2\pi i x} - e^{2\pi i y}| < 2 \pi |x-y|$, $A(k, \epsilon) = \{x: |e^{2\pi i k x} - 1| \le \epsilon\}$ contains an open interval of length $\epsilon/(2\pi^2 k)$
centred at each $j/k$ for integer $j$, $0 < j < k$.  If $k' = m k$ for positive integer $m > 4 \pi^2/\epsilon$, then each of these intervals contains at least two rationals of the form $j/k'$ for integer $k$, and thus (if $\epsilon'$ is small enough, two disjoint
intervals of $A(k', \epsilon')$.  So we can construct an increasing sequence $k_j$ of positive integers, a sequence $\epsilon_j \to 0+$, and a nested sequence $E_j$ of compact sets, such that $E_j$ is the union of $2^j$ intervals in $A(k_j,\epsilon_j)$.
   The intersection of the $E_j$ is then an uncountable Cantor-type set
on which $e^{2\pi i k_j x} \to 1$. 
A: I have an answer to my second question: $C$ is of Lebesgue measure zero.
By the Dominated Convergence Theorem and the Riemann-Lebsgue Lemma, we have
$$ \mu(C) = \int_\mathbb{R} \chi_C(t)\cdot 1dt = \lim_{n\to\infty}\int_\mathbb{R}\chi_C(t)\cdot e^{2\pi ik_nt}dt = 0,$$
where $\chi_C$ is the indicator function of $C$.
