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For $epsilon > 0$, is there always $n,m in mathbbN$ so that $e^n$ is $epsilon$-close to $m$? [closed]

1

$begingroup$

I don't have much to go off of, so I can't demonstrate any attempts here. I just want to know if there has been any answer or partial answer to this question.

analysis transcendental-numbers

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edited Mar 23 at 2:15

Fred

asked Mar 23 at 0:52

FredFred

62659

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closed as off-topic by Saad, Eevee Trainer, Lee David Chung Lin, max_zorn, Javi Mar 23 at 10:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Eevee Trainer, Lee David Chung Lin, max_zorn, Javi

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1

$begingroup$

I don't have much to go off of, so I can't demonstrate any attempts here. I just want to know if there has been any answer or partial answer to this question.

analysis transcendental-numbers

share|cite|improve this question

edited Mar 23 at 2:15

Fred

asked Mar 23 at 0:52

FredFred

62659

$endgroup$

closed as off-topic by Saad, Eevee Trainer, Lee David Chung Lin, max_zorn, Javi Mar 23 at 10:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, Eevee Trainer, Lee David Chung Lin, max_zorn, Javi

If this question can be reworded to fit the rules in the help center, please edit the question.

add a comment | 

$begingroup$

I don't have much to go off of, so I can't demonstrate any attempts here. I just want to know if there has been any answer or partial answer to this question.

analysis transcendental-numbers

share|cite|improve this question

edited Mar 23 at 2:15

Fred

asked Mar 23 at 0:52

FredFred

62659

$endgroup$

share|cite|improve this question

edited Mar 23 at 2:15

Fred

asked Mar 23 at 0:52

FredFred

62659

share|cite|improve this question

edited Mar 23 at 2:15

Fred

asked Mar 23 at 0:52

FredFred

62659

asked Mar 23 at 0:52

FredFred

62659

asked Mar 23 at 0:52

FredFred

62659

1 Answer
1

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5

$begingroup$

You can make $e^a$ exactly equal to any positive number $y$ (integer or not) by setting
$a = ln y$. You can get as close as you like with rational $a$.

I suspect that $e^n$ comes arbitrarily close to an integer for integral $n$, but have not been able to find a reference. Weyl's theorem implies that the integral multiples of $e$ (or any other irrational number) are equidistributed modulo $1$.

The powers of $e$ probably are, but caution is called for. There are irrational numbers $gamma$ whose powers are not equidistributed.
(https://math.unm.edu/~crisp/courses/wavelets/fall13/wavelet-weyl-report2.pdf)

share|cite|improve this answer

edited Mar 23 at 2:32

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ah now I feel silly. Of course... the integer values of $a$ are interesting. Thank you.
  $endgroup$
  – Fred
  Mar 23 at 1:11
  

  
  
  
  

add a comment | 

5

$begingroup$

You can make $e^a$ exactly equal to any positive number $y$ (integer or not) by setting
$a = ln y$. You can get as close as you like with rational $a$.

I suspect that $e^n$ comes arbitrarily close to an integer for integral $n$, but have not been able to find a reference. Weyl's theorem implies that the integral multiples of $e$ (or any other irrational number) are equidistributed modulo $1$.

The powers of $e$ probably are, but caution is called for. There are irrational numbers $gamma$ whose powers are not equidistributed.
(https://math.unm.edu/~crisp/courses/wavelets/fall13/wavelet-weyl-report2.pdf)

share|cite|improve this answer

edited Mar 23 at 2:32

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ah now I feel silly. Of course... the integer values of $a$ are interesting. Thank you.
  $endgroup$
  – Fred
  Mar 23 at 1:11
  

  
  
  
  

add a comment | 

5

$begingroup$

You can make $e^a$ exactly equal to any positive number $y$ (integer or not) by setting
$a = ln y$. You can get as close as you like with rational $a$.

I suspect that $e^n$ comes arbitrarily close to an integer for integral $n$, but have not been able to find a reference. Weyl's theorem implies that the integral multiples of $e$ (or any other irrational number) are equidistributed modulo $1$.

The powers of $e$ probably are, but caution is called for. There are irrational numbers $gamma$ whose powers are not equidistributed.
(https://math.unm.edu/~crisp/courses/wavelets/fall13/wavelet-weyl-report2.pdf)

share|cite|improve this answer

edited Mar 23 at 2:32

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ah now I feel silly. Of course... the integer values of $a$ are interesting. Thank you.
  $endgroup$
  – Fred
  Mar 23 at 1:11
  

  
  
  
  

add a comment | 

$begingroup$

You can make $e^a$ exactly equal to any positive number $y$ (integer or not) by setting
$a = ln y$. You can get as close as you like with rational $a$.

I suspect that $e^n$ comes arbitrarily close to an integer for integral $n$, but have not been able to find a reference. Weyl's theorem implies that the integral multiples of $e$ (or any other irrational number) are equidistributed modulo $1$.

The powers of $e$ probably are, but caution is called for. There are irrational numbers $gamma$ whose powers are not equidistributed.
(https://math.unm.edu/~crisp/courses/wavelets/fall13/wavelet-weyl-report2.pdf)

share|cite|improve this answer

edited Mar 23 at 2:32

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

$endgroup$

share|cite|improve this answer

edited Mar 23 at 2:32

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120

answered Mar 23 at 0:58

Ethan BolkerEthan Bolker

45.9k553120