Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate where $F_1(n)\neq F_2(n)$, then $d(F_1, F_2) = \exp[-\delta(F_1, F_2)]$ defines a metric on the space $\mathcal{F}_N$. One can observe that $d$ is an ultra-metric because, for each $F_1, F_2, F_3\in\mathcal{F}_N, d(F_1, F_3)\leq \max\Big[ d(F_1, F_2), d(F_2, F_3) \Big]$. Now fix $r\in(\frac{1}{2}, 1)$. My questions are:

1. Will there exists a finite set of positive integers $M$ such that $\sum_{i\in M}r^i = 1$? Does the existence of such a finite set depend on $r$?

2. Notice that we can view $\mathcal{F}_N$ as the family of all infinite subset of $\mathbb{N}$ (viewing each element in $\mathcal{F}_N$ as a sequence is more compatible to the given metric). For the fix $r\in(\frac{1}{2}, 1)$, one can observe that the set $\Big\{ \sum_{i\in F}r^i\,\vert\, F\in\mathcal{F}_N \Big\}$ is bounded below by $0$ (which is also the infimum) and bounded above by $\frac{r}{1-r}$. Is the following function surjective?

$$ S_r:\mathcal{F}_N\rightarrow (0, \frac{r}{1-r}], \hspace{0.5cm} F\mapsto \sum_{i\in F}r^i $$ In particular, I wonder, when part 1) is not true for all $r\in(\frac{1}{2}, 1)$, if it is true that, for each $r\in(\frac{1}{2}, 1)$, I can always find $F_r\in\mathcal{F}_N$ such that $S_r(F_r)=1$. One can easily check that $S_r$ is continuous for each $r\in(\frac{1}{2}, 1)$. If I instead let $\mathcal{F}_N$ be the set of all non-negative integers, then I suppose $S_{\frac{1}{n}}$ will be surjective for each $n\in\mathbb{N}$; however, under this assumption, I do not know if $S_r$ will be surjective even when $r\neq\frac{1}{n}$ for each $n\in\mathbb{N}$, but my current biggest concern is on the those two questions above.

Any hints or thoughts will be appreciated! The same question is posted in MS and I would like thank Ryszard Sszwarc for his help with this question. He proved that numbers that meet the condition in part 1) must be algebraic and irrational. However, for a fixed algebraic and irrational number, whether or not a necessary condition that guarantees the existence of such a finite set exists remain unclear.

Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate where $F_1(n)\neq F_2(n)$, then $d(F_1, F_2) = \exp[-\delta(F_1, F_2)]$ defines a metric on the space $\mathcal{F}_N$. One can observe that $d$ is an ultra-metric because, for each $F_1, F_2, F_3\in\mathcal{F}_N$, $$ d(F_1, F_3)\leq \max\Big[ d(F_1, F_2), d(F_2, F_3) \Big]. $$ Now fix $r\in\big(\frac{1}{2}, 1\big)$. My questions are:

1. Will there exists a finite set of positive integers $M$ such that $\sum_{i\in M}r^i = 1$? Does the existence of such a finite set depend on $r$?

2. Notice that we can view $\mathcal{F}_N$ as the family of all infinite subset of $\mathbb{N}$ (viewing each element in $\mathcal{F}_N$ as a sequence is more compatible to the given metric). For the fix $r\in\big(\frac{1}{2}, 1\big)$, one can observe that the set $\Big\{ \sum_{i\in F}r^i\,\vert\, F\in\mathcal{F}_N \Big\}$ is bounded below by $0$ (which is also the infimum) and bounded above by $\frac{r}{1-r}$. Is the following function surjective? $$ S_r:\mathcal{F}_N\rightarrow \Big(0, \frac{r}{1-r}\Big], \quad F\mapsto \sum_{i\in F}r^i $$ In particular, I wonder, when part 1) is not true for all $r\in\big(\frac{1}{2}, 1\big)$, if it is true that, for each $r\in\big(\frac{1}{2}, 1\big)$, I can always find $F_r\in\mathcal{F}_N$ such that $S_r(F_r)=1$. One can easily check that $S_r$ is continuous for each $r\in\big(\frac{1}{2}, 1\big)$. If I instead let $\mathcal{F}_N$ be the set of all non-negative integers, then I suppose $S_{\frac{1}{n}}$ will be surjective for each $n\in\mathbb{N}$; however, under this assumption, I do not know if $S_r$ will be surjective even when $r\neq\frac{1}{n}$ for each $n\in\mathbb{N}$, but my current biggest concern is on the those two questions above.

Any hints or thoughts will be appreciated! The same question is posted in MS and I would like thank Ryszard Sszwarc for his help with this question. He proved that numbers that meet the condition in part 1) must be algebraic and irrational. However, for a fixed algebraic and irrational number, whether or not a necessary condition that guarantees the existence of such a finite set exists remain unclear.

Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate where $F_1(n)\neq F_2(n)$, then $d(F_1, F_2) = \exp[-\delta(F_1, F_2)]$ defines a metric on the space $\mathcal{F}_N$. One can observe that $d$ is an ultra-metric because, for each $F_1, F_2, F_3\in\mathcal{F}_N, d(F_1, F_3)\leq \max\Big[ d(F_1, F_2), d(F_2, F_3) \Big]$. Now fix $r\in(\frac{1}{2}, 1)$. My questions are:

1. Will there exists a finite set of positive integers $M$ such that $\sum_{i\in M}r^i = 1$? Does the existence of such a finite set depend on $r$?

2. Notice that we can view $\mathcal{F}_N$ as the family of all infinite subset of $\mathbb{N}$ (viewing each element in $\mathcal{F}_N$ as a sequence is more compatible to the given metric). For the fix $r\in(\frac{1}{2}, 1)$, one can observe that the set $\Big\{ \sum_{i\in F}r^i\,\vert\, F\in\mathcal{F}_N \Big\}$ is bounded below by $0$ (which is also the infimum) and bounded above by $\frac{r}{1-r}$. Is the following function surjective?

$$ S_r:\mathcal{F}_N\rightarrow (0, \frac{r}{1-r}], \hspace{0.5cm} F\mapsto \sum_{i\in F}r^i $$ In particular, I wonder, when part 1) is not true for all $r\in(\frac{1}{2}, 1)$, if it is true that, for each $r\in(\frac{1}{2}, 1)$, I can always find $F_r\in\mathcal{F}_N$ such that $S_r(F_r)=1$. One can easily check that $S_r$ is continuous for each $r\in(\frac{1}{2}, 1)$. If I instead let $\mathcal{F}_N$ be the set of all non-negative integers, then I suppose $S_{\frac{1}{n}}$ will be surjective for each $n\in\mathbb{N}$; however, under this assumption, I do not know if $S_r$ will be surjective even when $r\neq\frac{1}{n}$ for each $n\in\mathbb{N}$, but my current biggest concern is on the those two questions above.

Any hints or thoughts will be appreciated! The same question is posted in MS and I would like thank Ryszard Sszwarc for his help with this question.

Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate where $F_1(n)\neq F_2(n)$, then $d(F_1, F_2) = \exp[-\delta(F_1, F_2)]$ defines a metric on the space $\mathcal{F}_N$. One can observe that $d$ is an ultra-metric because, for each $F_1, F_2, F_3\in\mathcal{F}_N, d(F_1, F_3)\leq \max\Big[ d(F_1, F_2), d(F_2, F_3) \Big]$. Now fix $r\in(\frac{1}{2}, 1)$. My questions are:

1. Will there exists a finite set of positive integers $M$ such that $\sum_{i\in M}r^i = 1$? Does the existence of such a finite set depend on $r$?

2. Notice that we can view $\mathcal{F}_N$ as the family of all infinite subset of $\mathbb{N}$ (viewing each element in $\mathcal{F}_N$ as a sequence is more compatible to the given metric). For the fix $r\in(\frac{1}{2}, 1)$, one can observe that the set $\Big\{ \sum_{i\in F}r^i\,\vert\, F\in\mathcal{F}_N \Big\}$ is bounded below by $0$ (which is also the infimum) and bounded above by $\frac{r}{1-r}$. Is the following function surjective?

$$ S_r:\mathcal{F}_N\rightarrow (0, \frac{r}{1-r}], \hspace{0.5cm} F\mapsto \sum_{i\in F}r^i $$ In particular, I wonder, when part 1) is not true for all $r\in(\frac{1}{2}, 1)$, if it is true that, for each $r\in(\frac{1}{2}, 1)$, I can always find $F_r\in\mathcal{F}_N$ such that $S_r(F_r)=1$. One can easily check that $S_r$ is continuous for each $r\in(\frac{1}{2}, 1)$. If I instead let $\mathcal{F}_N$ be the set of all non-negative integers, then I suppose $S_{\frac{1}{n}}$ will be surjective for each $n\in\mathbb{N}$; however, under this assumption, I do not know if $S_r$ will be surjective even when $r\neq\frac{1}{n}$ for each $n\in\mathbb{N}$, but my current biggest concern is on the those two questions above.

Any hints or thoughts will be appreciated! The same question is posted in MS and I would like thank Ryszard Sszwarc for his help with this question. He proved that numbers that meet the condition in part 1) must be algebraic and irrational. However, for a fixed algebraic and irrational number, whether or not a necessary condition that guarantees the existence of such a finite set exists remain unclear.