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-1

$begingroup$

I know the following statement is true for $k=0$.

Let $epsilon>0$ and $pi$ be irrational. The function $(cdot)$ denotes modulo $pi$. Then $forall kin mathbbNexists k_iin [k+1,cdots,k+lceilfracpiepsilonrceil]$ so that $(k_i)leqepsilon$ or $(k_i)geqpi-epsilon$

I found a proof for $k=0$ and a python-script told me that this statement is not true in general I find this somewhat counter intuitive.

I found a solution with numbers in $[k,k+klceilfracpiepsilonrceil]$ can i do better?

number-theory irrational-numbers

share|cite|improve this question

edited 2 days ago

A. P

asked 2 days ago

A. PA. P

1065

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is the question exactly?
  $endgroup$
  – Jack M
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I did some more digging and found im acctaually looking for upper bounds for the return time (to an interval of a given length) of an irrational rotation.
  $endgroup$
  – A. P
  19 hours ago
  

  
  
  
  

add a comment | 

-1

$begingroup$

I know the following statement is true for $k=0$.

Let $epsilon>0$ and $pi$ be irrational. The function $(cdot)$ denotes modulo $pi$. Then $forall kin mathbbNexists k_iin [k+1,cdots,k+lceilfracpiepsilonrceil]$ so that $(k_i)leqepsilon$ or $(k_i)geqpi-epsilon$

I found a proof for $k=0$ and a python-script told me that this statement is not true in general I find this somewhat counter intuitive.

I found a solution with numbers in $[k,k+klceilfracpiepsilonrceil]$ can i do better?

number-theory irrational-numbers

share|cite|improve this question

edited 2 days ago

A. P

asked 2 days ago

A. PA. P

1065

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What is the question exactly?
  $endgroup$
  – Jack M
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I did some more digging and found im acctaually looking for upper bounds for the return time (to an interval of a given length) of an irrational rotation.
  $endgroup$
  – A. P
  19 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I know the following statement is true for $k=0$.

Let $epsilon>0$ and $pi$ be irrational. The function $(cdot)$ denotes modulo $pi$. Then $forall kin mathbbNexists k_iin [k+1,cdots,k+lceilfracpiepsilonrceil]$ so that $(k_i)leqepsilon$ or $(k_i)geqpi-epsilon$

I found a proof for $k=0$ and a python-script told me that this statement is not true in general I find this somewhat counter intuitive.

I found a solution with numbers in $[k,k+klceilfracpiepsilonrceil]$ can i do better?

number-theory irrational-numbers

share|cite|improve this question

edited 2 days ago

A. P

asked 2 days ago

A. PA. P

1065

$endgroup$

share|cite|improve this question

edited 2 days ago

A. P

asked 2 days ago

A. PA. P

1065

share|cite|improve this question

edited 2 days ago

A. P

asked 2 days ago

A. PA. P

1065

asked 2 days ago

A. PA. P

1065

asked 2 days ago

A. PA. P

1065