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Dual of the Weil norm proof: Difference between revisions

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(Archived from Facebook, originally a post by Mike Battaglia addressing Gene Smith)
{{Archive}}


OK, so I proved the conjecture. The dual to the weil norm is max(<V 0|) - min(<V 0|). I'll formulate monzos as column vectors and vals as row vectors below.
(Archived from Facebook, originally a post by [[Mike Battaglia]] addressing [[Gene Smith]])
 
OK, so I proved the conjecture. The dual to the weil norm is max(<V 0|) - min(<V 0|). I'll formulate [[monzo]]s as column vectors and [[val]]s as row vectors below.


Let M be the space of p-limit real monzos with the Weil norm on them. The Weil norm of a weighted monzo |a b c ...> is 1/2*(|a| + |b| + |c| + ... + |a+b+c+...|), giving you log(max(n,d)).
Let M be the space of p-limit real monzos with the Weil norm on them. The Weil norm of a weighted monzo |a b c ...> is 1/2*(|a| + |b| + |c| + ... + |a+b+c+...|), giving you log(max(n,d)).
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which is a diagonal matrix with all .5's on the diagonal, followed by an extra row at the bottom of all .5's. I will call this matrix Ξ. So if you take any monzo m, L1(Ξm) = Weil(m).
which is a diagonal matrix with all .5's on the diagonal, followed by an extra row at the bottom of all .5's. I will call this matrix Ξ. So if you take any monzo m, L1(Ξm) = Weil(m).


Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of p-limit augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.
Since I called the original monzo space M, I will call this new, larger space M°; it is the L1 normed space of [[p-limit]] augmented monzos that have one extra coordinate tacked on the end. Ξ can be thought of as taking monzos in M to a subspace of M° - the subspace where the last coordinate is the sum of the others. Thus, you can see that the Weil norm itself can be thought of as the restriction of the L1 norm on M° to this subspace. By an abuse of notation, I will also call this subspace M.


Now, let's see if we can use this to figure out what the dual norm on vals looks like.
Now, let's see if we can use this to figure out what the dual norm on vals looks like.
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To do so, first we note that for any val v = <x y z ...|, the augmented val v° = <2x 2y 2z ...; 0| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression
To do so, first we note that for any val v = <x y z ...|, the augmented val v° = <2x 2y 2z ...; 0| with 0 tacked on at the end is in its preimage. So the norm on v is given by minimizing the expression


Linf(v° + kj°)
Linf(v° + rj°)


for k in R. Or, in other words, we want to minimize
for r in R. Or, in other words, we want to minimize


Linf(<2x 2y 2z ...; 0| + <k k k...; -k|)
Linf(<2x 2y 2z ...; 0| + <r r r...; -r|)


We can rewrite the above as follows
We can rewrite the above as follows


inf_k max(|2x+k|, |2y+k|, |2z+k|, ..., |k|)
inf_r max(|2x+r|, |2y+r|, |2z+r|, ..., |r|)


OK, so what's the k that minimizes that? It turns out to be the k such that
OK, so what's the r that minimizes that? It turns out to be the r such that


max(2x+k,2y+k,...,k) = -min(2x+k,2y+k,...,k)
max(2x+r,2y+r,...,r) = -min(2x+r,2y+r,...,r)


If k diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.
If r diverges from that in either direction, either the max coordinate increases or the min coordinate decreases, both of which have the effect of increasing the absolute value of that coordinate and hence the overall max.


Put differently, we want to take the range of <2x 2y 2z ...; 0| and set k so that this range is centered around zero.
Put differently, we want to take the range of <2x 2y 2z ...; 0| and set r so that this range is centered around zero.


But now we're done. We don't need to worry about finding k. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.
But now we're done. We don't need to worry about finding r. If the range is centered around zero, then max(abs(···)) over all coordinates is going to be half the range, and the above proves that this is going to be the minimized-max that we were searching for.


So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that
So finally, we prove that the quotient norm on v° is half its range, e.g. (max(v°)-min(v°))/2. And since we ended up doubling all of the coefficients of v when we translated it to v°, the division by 2 cancels that out, proving the original conjecture that
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QED.
QED.
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