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LEMMA 24: Let H be a subgroup of Q , and f : H + Q , an additive function. Assume that x E Hand 1x1, < 1 imply I f(x)lp < 1. Then there exist a positive b andap-adic number LX such that,for each x in H, 1f ( x ) - a x / , < b. In the proof it will be assumed that H i s dense in Q,. The obvious modifications should be made in the other case. (a) 1x1, > 1 implies ( f ( x ) J , < 1x1,. In fact, if pk = ]xip,then pkx E Z , and since f is additive, f ( p k x ) = pkf(x) (k 1 since lxlp > 1).

I f 8 is algebraic, if181, > 1 and i f ~ , ( B ~tends ) to zero, 8 is a Pisot-Chabauty number. I f 8 is a p-adic number, if 101, > 1 and if, for some p-adic number 1# 0 , x : (Q, (Mk))' < co,8 is a PisotChabauty number. Assume 8 is a Pisot-Chabauty number. , on(@ are integers over the ring Z,. Since 8 is integral over A,, O k ( ~ ~ ( 8 ) ) ~ + ( ~ ~ ( 8E)A,) ~for each positive k. Hence ( ~ ~ ( 8 ) ) ~ ( ~ ~ ( 8 ) ) ~ + + + + + 66 PROBLEMS I N LOCALLY COMPACT ABELIAN GROUPS PROBLEMS I N LOCALLY COMPACT ABELIAN GROUPS belongs to Q, and is integral over Z,.

Let V , = V(2-"). Then V , = - V, and V,,, + V,,, c V,. (V,),, is a sequence of neighborhoods of 0 in E defining topological group structure on E. Let F be the corresponding topology on Eand 9 the topology of the compact group R A . , LEMMA 14: For each p 3 1, Y and Finduce the same topology on V,. Proof: We write M , for M(2-"). SinceA is harmonious, M, is relatively dense and there exists a finite subset F, of G A such that % F, + M, = GA. We have h(M,) c V , and, since % is compact, so is V , + h(%) h(F,) which contains h ( G A )and is thus the whole of R A .

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