Why Exactly 3 Involutions Are Used in The Proof of Roger Heath-Brown?

Question

In the book "Proofs from THE BOOK" (click here, on page 20)by Martin Aigner and Günter M. Ziegler, a proof of representing number as sum of two square is given. The proof is due to Roger Heath-Brown (1971,appeared in 1984). The complete proof is given here -

Why exactly 3 involutions ($f,g, h$) are used in the proof? How we know more or less than 3 involution will not help/ is not required?

Related question is, why only $U, T$ are used? How we know $U, T$ covers $S$ completely, i.e. How can we prove $S= U \bigcup T$?

Answer 1

For your question

How we know more or less than 3 involution will not help/ is not required?

there is nothing in the proof that indicates that any involutions at all will be required in any other proofs of the theorem let alone the exact number of them. There are several other proofs of the theorem that do not use involutions. Read the Wikipedia article Fermat's theorem on sums of two squares for details about these other proofs. In fact, Zagier's proof is a simplification of Heath-Brown's proof and only uses two involutions.

For your question

How we know $U, T$ covers $S$ completely?

The proof does not require that they do so and this situation may or may not be true in any particular case.