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Irnlp < E. , ti). , tn and L , is the lattice defined in the first part of the proof. Since L , is relatively dense, we can solve I Y - t i Y , - .. , rn (a depends only on Y , , .. , Yn and 8 ) . Therefore D is relatively dense. To get the cor,verse result, the following lemma is needed: + SO + which gives Lemma 24. Returning to Proposition 10, let D,,for each a in Q,, be the set of all Since D is relatively dense, d = ( x , Y ) in Q , x Rn, such that 1x1, < D,is relatively dense in aZ, x Rnprovided lalpissufficientlylarge.

Our standard subset A of K was defined to be the set of all projections onto K = k, of all d E k whose projections onto k i belonged to the compact set n,,,D,. Hence A is a model. The proof of Theorem IX in the general case may be found in [89]. + 14. 1. a. group andA a subset of G. We say that Ais discrete if for each ilinA there exists a neighbourhood V of A in G whose intersection with A contains only A. We say that A is relatively dense if there exists a compact subset K of G such that any translate of K intersects A.

Group; we obtain a result similar to Theorem IV. a. a. group to Rnand we introduce the finite set F. 2. More general definition of models and the first part of Theorem I V rl'~A,,+l + K + + Q (2n 2)-I and consider a translate y, Vn of Vn; taking z = Xyl(4~+l), we can find a y, in y, Wn+, such that I~,,(il;+,) - 11 1/(2 (n 1)). Thus y, Vn+ c y, Vn and we have IX~(X,+ ,) - 1I < + + + , < l/(n + 1) for all y running in y, + + + Vn+l. 5. 1. The previous construction provides examples of lacunary stable harmonious sets.

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