By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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**Extra resources for A selection of problems in the theory of numbers **

**Example text**

Proof. 37) as v → 0. 18)). 37). 37), Sε is strictly contained between the spheres of radii ε(2π/c2 )1/2 and ε(2π/c1 )1/2 . Moreover, Sε is closed because S is continuous. 37). 26. 38). We have preferred, however, to give the general argument. v¯ Tv¯ Sε O· Fig. 7. The sphere-like manifold Sε . Finally, we define the functional I .. = (λ(v), v) . 27. A critical point v¯ ∈ Sε of I .. Sε → R is a critical point of (λ(v), ¯ ·).. Br (0) ⊂ V → R (with fixed period λ(v)). = v¯ + w(λ(v), ¯ v) ¯ is a 2πλ(v)-periodic ¯ solution of (H S) with energy H ( y¯ ) = ε2 .

Proof. We claim that if γ ∈ (0, 1/4], the set Wγ contains uncountably many irrational numbers ω such that its continued fraction expansion is ω = [1, a1 , a2 , . = 1 + for every a1 ∈ N and ai ∈ {1, 2} , 1 a1 + ∀i ≥ 2 1 a2 + . . 17) 38 2 Infinite Dimension (we refer to Appendix D for the basic notions about continued fractions). All these numbers ω belong to the open neighborhood (1, 1 + a1−1 ), and for a1 → +∞, they accumulate to ω = 1. Let us prove the previous claim. Take any pairs of integers (l, j) ∈ N × N, l = j.

Then v 1 · · · v 2k+1 = 0 . 48) In particular, if v ∈ V then v 2k ∈ W . Proof. 9). 48). 29). 25) = O( wε 2 ) = O(ε2 γ −2 ) . 25) = O(εγ −1 wε ) = O(ε2 γ −2 ) . 29), ε (v) ω2 − 1 v 4 1 v = −ε2 2 = 2 H1 2 H1 + ε2 − a2 2 a2 2 p v p L −1 ω v + εR3 p −1 v p L −1 ω v − ε R3 , having set the frequency–amplitude relation ω2 − 1 = −ε2 . 52) where ¤−1 .. 53) satisfies Rε (v) , (∇Rε (v), v) = O(εγ −2 ) . 54). 53) satisfies an even better estimate. Develop in Fourier series v p =.. 18). 51), |ω2l 2 − j 2 | ≥ γ /2, |l 2 − j 2 | ≥ 1.