Pillars of Transcendental Number Theory by Saradha Natarajan & Ravindranathan Thangadurai

Author:Saradha Natarajan & Ravindranathan Thangadurai
Language: eng
Format: epub
ISBN: 9789811541551
Publisher: Springer Singapore

and

Then

(5.24)

and

(5.25)

From the definition of and (5.24), it is easy to see that

(5.26)

Step 5. An upper bound for

By (5.20), assumption (5.5) and (5.25), we get

Let us now take Then Step 4 is valid and also

(5.27)

Step 6. Gap principle

For any two reduced rationals and with satisfying (5.5) we see that

which gives

In other words, denominators of such rationals are far apart. This phenomenon is also described as good approximations repel one another.

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