Let $(i_{n})$ be a strictly increasing sequence of natural numbers, $(v_{n})$ be an unbounded sequences of natural numbers and $M\geq 1$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all irrational numbers $\alpha=[a_{1}, a_{2},...,a_{s},...)\in (0,1)$ which is $a_{i_{n}}=v_{n}$ and $a_{s}\leq M$ for any $s\in \mathbb{N}\setminus \{i_{n}, n=1,2,...\}.$

My question related to the Lebesgue measure of this set. Of course this measure depends on exact form of sequences $i_{n}$ and $v_{n}.$ For example in my cases:

1. What is the Lebesgue measure of $\mathcal{I}(n^{4}, n^{6}, M)$?

2. What is the Lebesgue measure of $\mathcal{I}(n^{3}, n^{2}, M)$?

Let $(i_{n})$ be a strictly increasing sequence of natural numbers, $(v_{n})$ be an unbounded sequences of natural numbers and $M\geq 2$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all irrational numbers $\alpha=[a_{1}, a_{2},...,a_{s},...)\in (0,1)$ which is $a_{i_{n}}\leq v_{n}$ and $a_{s}\leq M$ for any $s\in \mathbb{N}\setminus \{i_{n}, n=1,2,...\}.$

My question related to the Lebesgue measure of this set. Of course this measure depends on exact form of sequences $i_{n}$ and $v_{n}.$ For example in my cases:

1. What is the Lebesgue measure of $\mathcal{I}(n^{4}, n^{6}, M)$?

2. What is the Lebesgue measure of $\mathcal{I}(n^{3}, n^{2}, M)$?

Let $(i_{n})$ be a strictly increasing sequence of natural numbers, $(v_{n})$ be an unbounded sequences of natural numbers and $M\geq 1$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all irrational numbers $\alpha=[a_{1}, a_{2},...,a_{s},...)\in (0,1)$ which is $a_{i_{n}}=v_{n}$ and $a_{s}\leq M$ for any $s\in \mathbb{N}\setminus \{i_{n}, n=1,2,...\}.$

My question related to the Lebesgue measure of this set. Of course this measure depends on exact form of sequences $i_{n}$ and $v_{n}.$ For example in my cases:

1. What is the Lebesgue measure of $\mathcal{I}(n, n^{2}, M)$?

2. What is the Lebesgue measure of $\mathcal{I}(n^{3}, n^{2}, M)$?

Let $(i_{n})$ be a strictly increasing sequence of natural numbers, $(v_{n})$ be an unbounded sequences of natural numbers and $M\geq 1$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all irrational numbers $\alpha=[a_{1}, a_{2},...,a_{s},...)\in (0,1)$ which is $a_{i_{n}}=v_{n}$ and $a_{s}\leq M$ for any $s\in \mathbb{N}\setminus \{i_{n}, n=1,2,...\}.$

My question related to the Lebesgue measure of this set. Of course this measure depends on exact form of sequences $i_{n}$ and $v_{n}.$ For example in my cases:

1. What is the Lebesgue measure of $\mathcal{I}(n^{4}, n^{6}, M)$?

2. What is the Lebesgue measure of $\mathcal{I}(n^{3}, n^{2}, M)$?