Thank You for your fast reply!
>>Thus A may be any integer between b/r-c/|r| and b/r+c/|r|
In the problem i meant the integer b is not fixed but depends on the choice of A (resp. a in my first post). That is, given real r and c>0, we are looking for all positive integers A for which A*r is not too far away from the nearest integer.
I did some experiments yesterday using Mupad. The result: the A's seem to have a regular behavior relative to their succesive differences. For example if we look at the first 500000 multiples of r=0.2703567032 (1*r,...,500000*r), so we get 10016 possible values for A with A*r at most 1/100 away from the nearest integer. I've numbered these A's and considered the succesive differences A_(i+1)-A_(i) and discovered that each of them takes one of the tree
possible values 122, 85 and 37 (notice that 122=85+37; this was true in all my considered examples except of them where fewer than 3 values were taken; there were never more than 3 possible values for the succesive differences, but it may be when we consider more then 500000 multiples). With the above r and c I get the first 100 succesive differences as below:
[37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37,
37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
122, 37, 37, 37, 37, 37, 37, 85, 37, 37, 37, 37, 37, 37, 122, 37,
37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37,
37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122,
37, 37, 37, 37]
while the 101 in the middle are
[37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
122, 37, 37, 37, 37, 37, 37, 85, 37, 37, 37, 37, 37, 37, 122, 37,
37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 37, 85, 37, 37,
37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37,
37, 37, 122, 37, 37]
and number 915-1015:
[37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 37,
85, 37, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37,
37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122,
37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 37, 122, 37, 37, 37,
37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37,
37, 37, 37, 37, 122, 37, 37, 37, 37, 37, 122, 37, 37, 37, 37, 37,
37, 85, 37, 37, 37].
The same setting with r:=0.8142678572 yields 1000 desirable A's among the first 500000 integers with the successive differences as follows:
Number 1-100:
[70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70,
70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70,
70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 70,
43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 70, 43,
27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 43, 27, 43,
27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 43]
Number 450-550:
[43, 27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 70, 43, 27, 43,
27, 43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27,
43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27,
43, 27, 43, 27, 43, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27,
43, 27, 43, 70, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43,
27, 43, 70, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43]
and number 899-999:
[27, 43, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43,
27, 43, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43,
70, 70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70,
70, 70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70,
70, 70, 70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70, 70,
70, 70, 70, 43, 27, 43, 27, 43, 27, 43, 27, 43, 27, 43, 70, 70].
Is there anything known about these A's in the literature? Is there a formula describing it? Any ideas?
Greetings,
eddik
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)