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Dirichlet's approximation theorem

Keywords: 
Dirichlet diophantine approximation
Synonym: 
Dirichlet approximation theorem
Type of Math Object: 
Theorem
Major Section: 
Reference

Mathematics Subject Classification

11J04 no label found

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Hi, I'm a new member and i am interested in the following closely related problem. Suppose you are given a real number r and a bound c>0. What does the set of all natural (or integral) numbers a satisfying |a*r-b| <= c for some integer b look like? Has this set any structure makes it easy to deal with it? Does anybody know if there are any good upper or lower bounds for the cardinality of the intersection of this set with a fixed interval [i1,i2]?
Would be great if anybody could help! Thanks in advance!

If r is distinct from zero, you can divide the inequality by |r|, getting
|A-b/r| <= c/|r|.
Thus A may be any integer between b/r-c/|r| and b/r+c/|r|.

Regards,
Jussi

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

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