linear combinations of of irrational number with integer coefficients.
$begingroup$
Let ${x_1, dots,x_k}$ be real numbers, under which conditions do I have that the set
$$
langle x_1, dots,x_krangle
=
{sum a_i x_i: a_i in mathbb{Z} }
$$
is discrete set?
Well, this is clearly the case if $x_i$'s are all integers. On the other hand, if I take $x_1=1$ and $x_2$ to be the Liouville Constant, I have a dense set. So what would be necessary and/or sufficient conditions?
real-analysis analysis
asked Dec 7 '18 at 8:00
KernelKernel
782521
$endgroup$
add a comment |
$begingroup$
Let ${x_1, dots,x_k}$ be real numbers, under which conditions do I have that the set
$$
langle x_1, dots,x_krangle
=
{sum a_i x_i: a_i in mathbb{Z} }
$$
is discrete set?
Well, this is clearly the case if $x_i$'s are all integers. On the other hand, if I take $x_1=1$ and $x_2$ to be the Liouville Constant, I have a dense set. So what would be necessary and/or sufficient conditions?
real-analysis analysis
asked Dec 7 '18 at 8:00
KernelKernel
782521
$endgroup$
add a comment |
$begingroup$
Let ${x_1, dots,x_k}$ be real numbers, under which conditions do I have that the set
$$
langle x_1, dots,x_krangle
=
{sum a_i x_i: a_i in mathbb{Z} }
$$
is discrete set?
Well, this is clearly the case if $x_i$'s are all integers. On the other hand, if I take $x_1=1$ and $x_2$ to be the Liouville Constant, I have a dense set. So what would be necessary and/or sufficient conditions?
real-analysis analysis
asked Dec 7 '18 at 8:00
KernelKernel
782521
$endgroup$
Let ${x_1, dots,x_k}$ be real numbers, under which conditions do I have that the set
$$
langle x_1, dots,x_krangle
=
{sum a_i x_i: a_i in mathbb{Z} }
$$
is discrete set?
Well, this is clearly the case if $x_i$'s are all integers. On the other hand, if I take $x_1=1$ and $x_2$ to be the Liouville Constant, I have a dense set. So what would be necessary and/or sufficient conditions?
real-analysis analysis
real-analysis analysis
asked Dec 7 '18 at 8:00
KernelKernel
782521
asked Dec 7 '18 at 8:00
KernelKernel
782521
asked Dec 7 '18 at 8:00
KernelKernel
782521
asked Dec 7 '18 at 8:00
KernelKernel
782521
asked Dec 7 '18 at 8:00
KernelKernel
782521
782521
add a comment |
add a comment |
1 Answer
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$begingroup$
If $frac x y$ is irrational then ${nx+my:n,m in mathbb Z}$ is dense. If $frac x y$ is rational then ${nx+my:n,m in mathbb Z}$ is discrete.
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
If $frac x y$ is irrational then ${nx+my:n,m in mathbb Z}$ is dense. If $frac x y$ is rational then ${nx+my:n,m in mathbb Z}$ is discrete.
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
add a comment |
$begingroup$
If $frac x y$ is irrational then ${nx+my:n,m in mathbb Z}$ is dense. If $frac x y$ is rational then ${nx+my:n,m in mathbb Z}$ is discrete.
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
add a comment |
$begingroup$
If $frac x y$ is irrational then ${nx+my:n,m in mathbb Z}$ is dense. If $frac x y$ is rational then ${nx+my:n,m in mathbb Z}$ is discrete.
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
$endgroup$
If $frac x y$ is irrational then ${nx+my:n,m in mathbb Z}$ is dense. If $frac x y$ is rational then ${nx+my:n,m in mathbb Z}$ is discrete.
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
answered Dec 7 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
59.2k42161
59.2k42161
add a comment |
add a comment |
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