Consistency, Smith style

I've used a measure which is related to the idea of consistency
proposed here, which I would like to explain.

Let w be an odd number, and let p_d <= w be the largest prime less
than or equal to w, and suppose that there are d primes p_i less than
or equal to l. Let h:G_p_d ->> Z be a homomorphism (more precisely
epimorphism, meaning onto) giving a p_d-limit et. Let {q_i} be the
set of rational numbers q_i >= 1 which are ratios of any two odd
numbers less than or equal to w, and let n = h(2). We define the w-
consistent goodness measure of h as

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))

In the most interesting cases, the homomorphism h will simply be the
homomorphism h_n obtained by rounding n*log_2(p_i) to the nearest
integer; this lets us define the w-consistent goodness of an integer
n, by setting

cons(w, n) = n^(1/d) * max(abs(n*log_2(q_i)) - h_n(q_i))

The d-th root of n is introduced because it is the appropriate
multiplier according to theorems relating to the simultaneous
Diophantine approximation of d independent numbers.

Here are two tables by way of example:

From 1 to 10000, 5-consistent measure cons(5, n) < 1

1 .736965595
2 .7439736471
3 .4245472985
4 .679700008
5 .8728704449
7 .6706891205
12 .5418300757
15 .8735997285
19 .5083949041
31 .8578063580
34 .6488389972
53 .4527427539
65 .7839449193
118 .4134352529
171 .6499654470
289 .9207676
441 .6622791
559 .9976240155
612 .5676032129
730 .6113208564
1171 .7597497149
1783 .5008376597
2513 .5396476355
4296 .2748910262
6809 .7979361504
8592 .7776946207

From 1 to 10000, 9-consistent measure cons(9, n) < 1.25

1 1.152003094
2 .9136193505
4 1.078956495
5 .7303891055
7 1.151607929
10 1.120372697
12 .8032252875
19 .9065721690
22 .9640922367
27 1.236074910
31 .9044163694
41 .8004237371
46 1.181094402
53 1.023130352
72 .9759757458
99 .8207791216
130 1.213753821
171 .3202177291
270 .7772103754
342 .8068974155
441 .8113651760
612 .9093921787
935 1.231274556
1106 1.219566982
1277 1.192780688
1848 1.206088177
2954 1.055468357
3125 .6018359509
3296 1.116123065
3566 1.147361792
6691 .9930572626
8539 1.219825812

--- In tuning-math@y..., genewardsmith@j... wrote:

> Let w be an odd number, and let p_d <= w be the largest prime less
> than or equal to w, and suppose that there are d primes p_i less
than
> or equal to l.

Sorry, an ascii "l" looks like the number "1", so I changed the l's
to w's but missed this one. It should be p_i less than or equal to w.

I'll have to look at this later, Gene. Sounds very interesting and
not unlike some things I've mentioned on the tuning list about 5
years ago when it was on the Mills server.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'll have to look at this later, Gene. Sounds very interesting and
> not unlike some things I've mentioned on the tuning list about 5
> years ago when it was on the Mills server.

Incidentally, if you or anyone else is interested in Maple routines
for functions such as cons(w, n) I could post them.