SHORT PROOFS OF THEOREMS OF LEKKERKERKER AND BALLIEU

For any irrational number let A() denote the set of all accumulation points of {z: z=q(q- p), p , q

and A() stands for the set of all the real accumulation points of {z: z q(q-p), p e q -{0}, p and q relatively prime}.Obviously, A() describes those Dirichlet approximation qualities which occur infinitely often.Furthermore, for any sequence (a) let H(a denote the set of all its limit points and for x e R n n and g > 0 set B(x,g) (x g, x + g).The main purpose of this paper is to gJ'e a simple proof of the following theorem of Lekkerkerker [i] (cf.also [2])" The set A() is discrete and does not contain zero if and only if is a quadratic irrational.
The proof of the sufficient part of the theorem mainly depends on the irre- ducible polynomial of , whereas the necessary part is a consequence of the relation between A() and the sequence (n) and simultaneously establishes the following theorem of Ballieu [3]: The set H( is finite and () is bounded if and only f is a quadratic n n irrational.
Finally for any quadratic irrational we shall show how to evaluate A() in an easy way.

BAS IC FORMULAS.
In this section we state the formulas used in the sequel.Let A /B denote n n the n-th convergent of [bo,bl,b2,... where An and Bn are relatively prime.Set O n Bn/Bn_ 1 and put 6n Bn(Bn-An)" Then the following formulas hold for all n e]O:  PROOF.Equation (2.1) is an easy consequence of the well known identity -An/Bn (-l)n/(Bn[Bnn+l_+ Bn_,l)),formula (2.2) can be derived from (2.1) when b + i/ and, finally (2 3) can using the identities n b + i/n+ I and O n n n 0n-i be obtained when combining (2.1) and (2.2).
(ii) Suppose that A() is discrete and 0 A().From equation (2.1) we can see that 16nI -< 1 for all n q0" Therefore and since all the numbers n are distinct, H(n) is a compact subset of A() and hence H(n is finite and 0 H(n).Now by (2.3) it is easy to see that H( n) is finite and (n) is bounded.Therefore, in order to complete the proof, it suffices to prove Ballieu's theorem.
(i) Suppose that ($n,) is bounded and H(n It follows from the identity n [bn'bn+l'''" that there exists a k lq such that bn < k for all n IN, and hence bn + i/k < n < bn + i i/(k + i).Therefore, the set H(n) n -is empty and we can find a number g > 0 such that th sets B(zv,g) are pairwise disjoint and contained in]R-Z.
Let l(z) denote the greatest integer not exceeding z and for z Z put Clearly, n+l I(n for all n IN0" Also the function I is continuous on H(n) therefore I(H(n) H(n) and we can find a 6, 0 < 6 < g, such that I(B(z,$)) = B(I(z),g) for all v {i m} There exists a number n O m such that n B(Zl,) and n B(z,) for all n > n Therefore, when o =I o writing (P) for the p-th composition map of , we obtain by induction that n + p e B(I(P)(zl)'6) for all p e0" Since (H(n)) H(n) and H(n is finite, O the sequence ((P)(zl)), p lq0, is periodic.From the identities bn l(n + p) (Zl)) we conclude that the sequence (b n + P 0' is periodic and thus, O by Lagrange's theorem, is a quadratic irrational. (ii) The other direction of Ballieu's theorem is an easy consequence of i. RI EDERLE Lagrange's theorem.

CONCLUDING REkRKS.
The inclusion in (3.3) is actually an equality.In order to prove this, we need the following well known theorem (cf.[4], p. 22-23): Let f(x,y) be an indefinite quadratic form with integer coefficients and let be one of its roots.Then for any pair (p,q) e Z x(-{0}) there are infinitely many relatively prime integers Pn' qn such that f(pn,qn f(p,q) for all n and nlm (qn p) O.
In fact, this result combined with ( 5) and ( 6), leads to the following: THEOREM.Suppose that is a quadratic irrational, say f(,l) 0 for some indefinite quadratic form f(x,y) with integer coefficients.Moreover, let E be the algebraic conjugate of .Then

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.