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While I was studying the function:

$$f_n(x)=\frac {\lfloor\ln(x^{n+1})\rfloor}{\lfloor\ln(x)\rfloor}-\frac{\lfloor\ln(x^{n})\rfloor}{\lfloor\ln(x)\rfloor}\quad\quad\quad\quad\quad\quad\quad\quad\quad\,(0)$$

I've notice that the values of $n$ for which the limit of $f$ as $x\rightarrow \frac{1}{2}$ which evaluate to a positive binary digit (an element of ${\{0,1}\}$) are equal a pair of complementary Beatty sequences generated by the cubic algebraic irrational number $3^{\frac{1}{3}}$, one for each of the two values as follows:

$${\{n \in \mathbb N :\lim _{x\rightarrow \frac{1}{2}}(f_{{n}} \left( x \right))=1 }\}={\{\mathfrak B_n(3^{\frac{1}{3}})}\} \, \quad\quad\quad\quad\quad\quad (1a)$$

$${\{n \in \mathbb N :\lim _{x\rightarrow \frac{1}{2}}(f_{{n}} \left( x \right))=0}\}={\Biggr\{\mathfrak B_n\Biggl(\frac{3^{\frac{1}{3}}}{3^{\frac{1}{3}}-1}\Biggr)}\Biggr\} \quad\quad\quad(1b)$$

$$\lim _{x\rightarrow \frac{1}{2}}(f_{{n}} \left( x \right))=\Biggl\lfloor\ln\Biggl(\frac{1}{2^{n}}\Biggr)\Biggr\rfloor-\Biggl\lfloor\ln\Biggl(\frac{1}{2^{n+1}}\Biggr)\Biggr\rfloor\quad\quad\quad\quad\quad\,(2)$$

Substitution of an element of the the sets defined by these Beatty sequences into (2) produce the the algebraic identities:

$$n \rightarrow\lfloor3^{1/3}n\rfloor \quad yields:$$ $$\lfloor\ln(2) (\lfloor 3^{1/3}n\rfloor+1)\rfloor-\lfloor\lfloor 3^{1/3}n\rfloor \ln(2)\rfloor=1 \quad\forall\, n \in \mathbb N\quad\,\,\,\,\,\quad\quad\quad\,\,\,\,\, (3a)$$

$$n \rightarrow\Biggl\lfloor \frac{3^{1/3}n}{3^{1/3}-1}\Biggr\rfloor\quad yields:$$

$$\Biggl\lfloor\ln(2) \Bigl(\Bigl\lfloor\frac{3^{1/3}n}{3^{1/3}-1}\Bigr\rfloor+1\Bigr)\Biggr\rfloor-\Biggl\lfloor\Bigl\lfloor\frac{3^{1/3}n}{3^{1/3}-1}\Bigr\rfloor \ln(2)\Biggr\rfloor=0 \quad\forall\, n \in \mathbb N\quad\,\,\, (3b)$$

So my questions are:

1) should this unusual result be considered another one of those odd coincidences that get categorized as recreational mathematics by which ever committee of people that decide what is legitimate and what is recreational?

2) If it is something that is of value to others, what other functions do these sequences appear to have number theoretic significance? presumably in the extension of the rational numbers to the algebraic irrational numbers so can I be given a reference to the theorem governing this case example please.

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    $\begingroup$ "legitimate" and "recreational" are not antonyms. $\endgroup$ – Gerry Myerson Jul 18 '18 at 12:49
  • $\begingroup$ So what would your suggestion be as a replacement for legitimate in the above context? $\endgroup$ – Adam Jul 18 '18 at 12:55
  • $\begingroup$ My suggestion would be a question mark after "coincidences" and delete the rest of the paragraph. $\endgroup$ – Gerry Myerson Jul 18 '18 at 12:59

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