BOP tuning: Difference between revisions

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This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, where this weighting is given by the dual L1 norm. Furthermore, this shows that the weighted error will always be obtained at a prime.

Technically, the supremum above also includes all vectors with arbitrary real coordinates, whereas we want to restrict to only those that have integer coordinates in the unweighted basis. However, since we divide by the norm, we note the supremum with all integer monzos is the same as the supremum with all monzos with rational coordinates, which is dense in the set of arbitrary real-valued vectors, so that the supremum will be the same - in particular, it is always found at a prime, which is an integer-coordinate anyway.

Now, if our weighting matrix is the usual <math>1/\log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that the tuning that minimizing the max Tenney-weighted error on the primes also minimizes the max Tenney-weighted error on all intervals. This is called the [[TOP tuning]]. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by

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 This tells us that the max weighted error on the primes has the property of also being the max weighted error on ''all'' monzos in the prime-limit, where this weighting is given by the dual L1 norm. Furthermore, this shows that the weighted error will always be obtained at a prime.
+Technically, the supremum above also includes all vectors with arbitrary real coordinates, whereas we want to restrict to only those that have integer coordinates in the unweighted basis. However, since we divide by the norm, we note the supremum with all integer monzos is the same as the supremum with all monzos with rational coordinates, which is dense in the set of arbitrary real-valued vectors, so that the supremum will be the same - in particular, it is always found at a prime, which is an integer-coordinate anyway.
 Now, if our weighting matrix is the usual <math>1/\log(p)</math> Tenney-weighting matrix, then the above is equivalent to [[Paul Erlich]]'s theorem that the tuning that minimizing the max Tenney-weighted error on the primes also minimizes the max Tenney-weighted error on all intervals. This is called the [[TOP tuning]]. However, if we instead change the weighting matrix to <math>1/p^s</math> instead, then our Linf norm will be dual to a different, somewhat unusual weighted L1 norm on monzos: the one where the weighting on the primes is given by <math>p^s</math>, and the weighting for an arbitrary monzo <math>m = |a\, b\, c\, ...\rangle</math> is given by