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Citable reference (book) for a variant of Kronecker's Approximation Theorem

A variant of Kronecker's approximation theorem?Weierstrass Approximation Theorem for continuous functions on open intervalQuestion about the Weierstrass approximation theoremReference on a variant of the Lagrange's four-square theoremArzela-Ascoli variant for $C^1$ functionsReference request: “mathematical systems for probability”Find $fracab in mathbbQ$ such that $ |,fracab - sqrt2|_3 < epsilon $Reference request: Federer-Besicovitch structure theoremApproximation to $1.05^1.02$ with Taylor's Theorem $T_2f(a;(x,y))$ in the point $a = (1,1)$some reference requests for Borel algebrasA variant of Kronecker's approximation theorem?

0

$begingroup$

Let $tau,sigmain(0,infty)$ with $fractausigmanotinmathbb Q$. A common version of Kronecker's Approximation Theorem is the following:

For each $xin mathbb R$ and $epsilon>0$, there are $m,ninmathbb N$ such that $|x+ntau-msigma|<epsilon$.

This answer to a recent question of mine provides the following variant:

For each $epsilon>0$, there is some $N(epsilon)inBbb N$ such that for all $xin mathbb [0,infty)$ there are $minmathbb N$ and $nin0,ldots,N(epsilon)$ such that $|x+ntau-msigma|<epsilon$.

The linked answer contains a (simple and correct) proof, so I have absolutely no questions about that.

My question is the following: Does anyone know of a citable reference (book) which contains this statement?

Thanks a lot in advance.

reference-request approximation diophantine-approximation

share|cite|improve this question

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

$endgroup$

add a comment | 

0

$begingroup$

Let $tau,sigmain(0,infty)$ with $fractausigmanotinmathbb Q$. A common version of Kronecker's Approximation Theorem is the following:

For each $xin mathbb R$ and $epsilon>0$, there are $m,ninmathbb N$ such that $|x+ntau-msigma|<epsilon$.

This answer to a recent question of mine provides the following variant:

For each $epsilon>0$, there is some $N(epsilon)inBbb N$ such that for all $xin mathbb [0,infty)$ there are $minmathbb N$ and $nin0,ldots,N(epsilon)$ such that $|x+ntau-msigma|<epsilon$.

The linked answer contains a (simple and correct) proof, so I have absolutely no questions about that.

My question is the following: Does anyone know of a citable reference (book) which contains this statement?

Thanks a lot in advance.

reference-request approximation diophantine-approximation

share|cite|improve this question

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

$endgroup$

add a comment | 

$begingroup$

Let $tau,sigmain(0,infty)$ with $fractausigmanotinmathbb Q$. A common version of Kronecker's Approximation Theorem is the following:

For each $xin mathbb R$ and $epsilon>0$, there are $m,ninmathbb N$ such that $|x+ntau-msigma|<epsilon$.

This answer to a recent question of mine provides the following variant:

For each $epsilon>0$, there is some $N(epsilon)inBbb N$ such that for all $xin mathbb [0,infty)$ there are $minmathbb N$ and $nin0,ldots,N(epsilon)$ such that $|x+ntau-msigma|<epsilon$.

The linked answer contains a (simple and correct) proof, so I have absolutely no questions about that.

My question is the following: Does anyone know of a citable reference (book) which contains this statement?

Thanks a lot in advance.

reference-request approximation diophantine-approximation

share|cite|improve this question

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

$endgroup$

share|cite|improve this question

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

share|cite|improve this question

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121

asked Mar 13 at 18:31

Mars PlasticMars Plastic

1,480121