S = {(t-n*a, beta*t-m*a), for t real, m, n integers} Apply the

This means, given (t

By replacing t = t

Setting t

This can be proved by applying the

T = {t

Thus, T is an infinite set of I, and by the thoerem referred above has a sequence {s

Considering the form of the elements of T, we can assume that d = s

The above figure illustrates the assertion proved. The line defined parametrically by t*(1,beta), when intercepts the boundary of the square at a point (a,x) or (x,a), it is continued in the same direction from the point lying on the opposite side i.e from (0,x) or (x,0). By repeating this procedure we create the above image representing a part of set S. If beta is irrational and we continue drawing, then the green lines will cover the whole area of the square.

Note that a similar argument prooves that for an angle w, with w/pi irrational the multiples n*w define a dense set on the unit circle.

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