The division of segment
Preliminary.
Let some number  be chosen, and one marks points {, {2, {3, {(n-1)on segment
[0,1].
If the contrary isn' t expressly stated, we assume that  is irrational. If  = p/q, we assume that n<q.
Thus, whatever  is, points do not coincide.
Thus, segment [0,1] is divided into n parts. Moreover let us assume that n>10 and 0.3<{}<0.7.
This limitation is irrelevant, we impose it to exclude some trivial effects for small numbers. It
means, in particular, that each part of division is less than {}.
Note also that if you replace  by n+ or n-, then you get the same parts. So when the question
concerns uniqueness, we suppose that 0<<1/2.
The content of the task is to investigate what parts we get and how they are located.
The tasks.
The ratio of the lengths of the longest and the shortest segments, we denote by L = L(, n).
A1. Let be the rational number,  = p/q. Prove that there exist n for which
L (n) = 1.
A2. Does there exist some other integer or rational k, k>1, for which one may state that for any
rational  there exist n such that L (n) = k ?
Now we reject the assumption that  is rational.
B1. Prove that for any n, there exist only 3 or less parts of different length. (Obviously, if  is
irrational, there are two or more different lengths).
For given  we call the number n "double number", if there are only two different lengths of
segments, and "triple number" if there are three of them.
B2. Prove that if is irrational, then there exist infinitely many double and infinitely many triple n.
B3. Given the irrational number ; n runs values n=1, 2, ..., m. Prove that if m tends to infinity, the
ratio of double numbers among them tends to zero.
Let φ (m) be a number of double numbers n among the numbers n=1, 2, ..., m. Rate the speed with
which the ratio φ (m)/m tends to zero as m tends to infinity.
B4. For what numbers  this ratio tends to zero maximally slowly? And what numbers  this ratio
tends to zero maximally quickly?
(You may simply give a few examples, provided that these examples will be convincing).
B5. Estimate from above and from below the number of double numbers
if 1 < n < 1.000.000. (preferably, of course, more or less accurate; estimations like "more than
three" are not accepted).
C1. Is there a number  , such that L =L(n) > 10 for all n > 9 ?
C2. Let n = 2.000.000. Can it happen that some interval [a, a + ½] contains more than 1.100.000
points? (Recall the assumption 0.3 < {} < 0.7).
C3. Does there exist , for which L takes
(a) the infinite,
(b) a finite number of different values, when n runs all the values from 10 to infinity.
Give examples of such  (if they exist).
C4. Find some sufficient conditions for L to take only the final (infinite) number of values. (If
possible - find necessary and sufficient condition; but you can also give some sufficient conditions).
D1. Prove that for any given irrational  there exists such a value A that L (n) =A more than
thousand times (for different n).
D2. Logically, there are three possible alternatives:
(*) for any  there exists such value A that L=A an infinite number of times,
(**) for any  and for any A, L(n) =A only for finite number on n,
(***) for some , the first variant is true, and for some the second.
Which of these alternatives is correct? If the third, for which  the first is true and for which – the
second one?
D3. Assume that for some  L takes both values A anb B at least once (for n>10).
Is it true that the assertions:
(*) L takes the value A only finite number of times and
(**) L takes the value B only finite number of times
are equivalent?