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If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here. A potential exception is the irrational number $x =0.10100100010000100000\cdots$ in base $b=2$.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

Context

We are dealing with what I call $b$-processes in my article about the Theory of Randomness, see here.

If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here. A potential exception is the irrational number $x =0.10100100010000100000\cdots$ in base $b=2$.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

Context

We are dealing with what I call $b$-processes in my article about the Theory of Randomness, see my book on chaotic dynamical systems. There is a follow up to this question, here.

If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here. A potential exception is the irrational number $x =0.10100100010000100000\cdots$.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

Context

We are dealing with what I call $b$-processes in my article about the Theory of Randomness, see here.

If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here. A potential exception is the irrational number $x =0.10100100010000100000\cdots$ in base $b=2$.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

Context

We are dealing with what I call $b$-processes in my article about the Theory of Randomness, see here.

If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

If $x$ is an irrational number and $b$ an integer, let's define

$g(x,k) = \mbox{Correl}(\{nx\},\{nb^kx\})$.

Here $k=1,2,\cdots$ is an integer. The brackets represent the fractional part function. The function $g$ is the empirical correlation computed over all $n=1,2, \cdots$. I asked whether one can prove that $g(x,k) = b^{-k}$. I also asked if there are some irrational numbers $x$ for which this equality is NOT true. I believe, based on my knowledge about mathematical chaos theory, that there are infinitely many exceptions, but these exceptions are extremely rare (among irrational numbers) and not easy to find. I initially posted the question on MSE, here. A potential exception is the irrational number $x =0.10100100010000100000\cdots$.

I eventually posted a partial answer myself, and along the way, found (using empirical evidence, no formal proof) that if $b=p/q$ where $p,q$ are strictly positive integers, and $\mbox{gcd}(p,q)=1$, then $g(x,1)=\frac{1}{pq}$. This can be used trivially to solve the general case $k>1$.

I am curious to see if someone might be interested in this problem, and provide more a formal answer, not based on empirical evidence only.

Definition of correlation (please read)

I was asked to clarify what I meant by correlation, since the system is deterministic and involves infinitely many terms, resulting in potential convergence issues. Let $X_n = \{nx\}, Y_n=\{nb^k x\}$ and then define the following quantities:

$E_m[X] = (\sum_{n=1}^m X_n)/m$, $E_m[Y] = (\sum_{n=1}^m Y_n)/m$, $\mbox{Prod}_m[X, Y]= (\sum_{n=1}^m X_n Y_n)/m$.

Similarly, define $\mbox{Var}_m[X]$ and $\mbox{Var}_m[Y]$. Finally define

$\mbox{Corr}_m[X,Y] = (\mbox{Prod}_m[X,Y]-E_m[X]E_m[Y])/\sqrt{\mbox{Var}_m[X]\mbox{Var}_m[Y]}$.

The correlation is the limit as $m\rightarrow\infty$ of $\mbox{Corr}_m[X, Y]$. The limit in questions almost always exists except in extreme cases (singularities) such as $b=2\sqrt{2}, x = \sqrt{2/2}$, and $k=1$.

Note that the sequences $X_n$ and $Y_n$ defined above are time series, related to Brownian motions (discrete version of derivatives of Brownian motions).

Interesting update

If $b$ is irrational, and $b^k x$ and $x$ are linearly independent over the set of rational numbers, then $g(x, k) = 0$. This fact seems trivial to me, and confirmed by empirical evidence, but could be extremely difficult to formally prove.

Context

We are dealing with what I call $b$-processes in my article about the Theory of Randomness, see here.