Harmonic entropy: Difference between revisions

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Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the **domain-integral approach**, works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the **complexity-normalization approach**, has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below.

For sets of basis rationals which are nowhere dense, the log-frequency spectrum can be divided up into **domains** assigned to the basis rationals. Each rational is assigned a domain with lower bound equal to the mediant of itself and its nearest lower neighbor, and likewise with upper bound equal to the mediant of itself and its nearest upper neighbor. If no such neighbor exists, ±∞ is used instead. Mathematically, this can be represented via the following expression:

In the case where the set of basis rationals consists of a finite set bounded by Tenney or Weil height, the resulting set of widths is conjectured to have interesting mathematical properties, leading to mathematically nice conceptual simplifications of the model. These simplifications are explained below.

It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff N as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to 1/sqrt(nd). While it's still an open conjecture that this pattern holds for arbitrarily large N, the assumption is sometimes made that this is the case, and hence that for these basis rational sets, 1/sqrt(nd) "approximations" to the width are sufficient to estimate domain-integral Harmonic Entropy. This modifies the expression for the p&lt;span style="font-size: 12px; vertical-align: sub;"&gt;d&lt;/span&gt;(b) as follows, noting that for the moment the "probabilities" won't sum to 1:

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Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the &lt;strong&gt;domain-integral approach&lt;/strong&gt;, works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the &lt;strong&gt;complexity-normalization approach&lt;/strong&gt;, has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below.&lt;br /&gt;

&lt;br /&gt;

  For sets of basis rationals which are nowhere dense, the log-frequency spectrum can be divided up into &lt;strong&gt;domains&lt;/strong&gt; assigned to the basis rationals. Each rational is assigned a domain with lower bound equal to the mediant of itself and its nearest lower neighbor, and likewise with upper bound equal to the mediant of itself and its nearest upper neighbor. If no such neighbor exists, ±∞ is used instead. Mathematically, this can be represented via the following expression:&lt;br /&gt;

&lt;br /&gt;

In the case where the set of basis rationals consists of a finite set bounded by Tenney or Weil height, the resulting set of widths is conjectured to have interesting mathematical properties, leading to mathematically nice conceptual simplifications of the model. These simplifications are explained below.&lt;br /&gt;

&lt;br /&gt;

  It has been noted empirically by Paul Erlich that, given all those rationals with Tenney height under some cutoff N as a basis set, that the domain widths for rationals sufficiently far from the cutoff seem to be proportional to 1/sqrt(nd). While it's still an open conjecture that this pattern holds for arbitrarily large N, the assumption is sometimes made that this is the case, and hence that for these basis rational sets, 1/sqrt(nd) &amp;quot;approximations&amp;quot; to the width are sufficient to estimate domain-integral Harmonic Entropy. This modifies the expression for the p&lt;span style="font-size: 12px; vertical-align: sub;"&gt;d&lt;/span&gt;(b) as follows, noting that for the moment the &amp;quot;probabilities&amp;quot; won't sum to 1:&lt;br /&gt;

&lt;br /&gt;