R 0C Z N IK I POLSKIEGO TOWARZYSTWA MATEMAT Y CZNE GO Séria I : PRACE MATEMATYCZNE X V (1971)
A N N A L E S SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)
E. W. C haney (California)
Note on Fourier-Stieltjes transforms of continuous and absolutely continuous measures
Let G be a compact abelian group and let Г be its character group.
Let M(G) be the Banach algebra of bounded Eadon measures on G.
In this note we shall consider mainly the case in which G is the circle group T and Г is therefore (canonically isomorphic to) the group Z of integers. The set of positive integers is denoted by Z +.
We shall construct a subset A of Z which has these two properties:
(i) If /л is in M{T) and if the Fourier-Stieltjes transform у of p va
nishes on the complement A' of A, then
1л is continuous (i.e., non-atomic).
(ii) There exists /л in M(T) such that p vanishes on A' and yet /л is not absolutely continuous.
1. D efinition . Let b be in Z+. A subset В of Z+ is said to be an arith
metic progression of type b if there exist Tc and n in Z such that h < n and В = {mb : m eZ+ , h < m < n}.
2. L emma . Let b be in Z+ and suppose that E is a subset of Z+ which contains arbitrarily long arithmetic progressions of type b. Then the set A — E и —E satisfies condition (ii).
Proof. We shall define by induction a sequence {%}£°=1 of positive integers. First choose nx in Z+ so that nxb is in E. Having chosen nx, n2, ...
. . . , n k we now choose nk+x so large that Ъ{пхА-п2-\- . . . + % ) < nk+x and so that E contains the arithmetic progression
{mb : m*Z+ , \m—nk+1\ < nx + n2+ . . . + nk}.
In the terminology of Hewitt and Zuckerman ([2], 3.1 and 3.4), the set of characters
x -> exp (ibnkx), h
eZ+ ,
is a dissociate set. By [2], 3.2, there exists y in M(T) such that
e 2 ^mfc2+ • • • + £то^%т ) = 2 w
148 R. W. C h a n e y
for all subsets {&х, h2, ..., Jcm} of Z+ and all sequences , ег, ..., em}
with values in the set {— 1 , 1 }; moreover, ^( 0 ) = 1 and ju(n) = 0 when
ever n is in Z but is not of the form
( 1 ) n = ег bnkj.+ e 2 bnkz -f ... + sm bnkm.
It follows at once from the Biemann-Lebesgue Lemma that ju is not absolutely continuous. It remains to show that [л(п) = 0 whenever n is not in A — E kj —E.
Suppose then that jù{n) Ф 0, for n in Z. Then n must be of the form
m —1 1
( 1 ) where h1< \ < ... < h m and ^ = ± 1 - Now | £/%- l < Л % and so
1= i 1 3= 1
fcm —1
£mbnkm- nqb < п ф £mbnkm-\- £ nqb.
3 = 1 3= 1
Since n is divisible by b it follows that n is in A. The proof of the lemma is complete.
3. Remark. It is known (see [3], p. 108, or [1], Example 1) that a measure ц in M (T) is continuous if and only if
lim
n->OQ
1 2 n + 1
П
£ i A w i 2
k = —n
= 0 .
4. Now we define E to be the set of all positive integers of the form nz—Tc, where n ranges over Z + and Jc = 0 , 1 , . . . , n — 1 . In view of 2 , A = E и — E satisfies condition (ii).
In order to prove that A also satisfies (i), we shall use 3. Given n in Z +, let c{n) denote the cardinality of the set E n {IceZ+ : Jc < n}. Now suppose that ц is in M (T) and that jx (n) = 0 whenever n is not in A.
For each n in Z +, we have П
(2) N l A W I ! < ( 2 c ( » ) + l ) i W I 2-
k — —n
m
For each m in Z + it is true that c(m3) — = \m{m-\-±). And,
3 = 1
to each positive integer n there corresponds a unique m in Z+ such that m 3 < n<z ( m-fl )3; clearly п~гс(п) < m_ 3 c((m-(-l)3) = fm_ 3 (m+l )(m-f- 2 ).
It now follows from (2) and from 3 that A satisfies condition (i).
5. Bern ark. Subsets of Г which satisfy both conditions (i) and (ii)
can be constructed for certain 0 -dimensional groups G. We shall not discuss
this in detail. Bather, we shall merely point out that the result in [ 2 ],
C o n t i n u o u s m e a s u r e s 149
4.4, can be used to construct sets which satisfy (ii) for G and that the result in [1], Example 5, can be used to show that certain of these sets satisfy (i); it should be observed here that Theorem 5 in [1] is a gener
alization of 3.
References
[1] E. H e w it t and K. R. S tr o m b e r g , A remarie on Fourier-Stieltjes transforms, An. Acad. Brasil. Ci. 34 (1962), p. 175-180.
[2] E. H e w it t and H. S. Z u ck erm a n , Singular measures with absolutely continuous convolution squares, Proc. Camb. Phil Soc. 62 (1966), p. 399-420.
[3] A. Z y g m u n d , Trigonometric series (Vol. I), Cambridge University Press, London 1959.
U N IV E R S IT Y OF CAL IFO R N IA , SAN TA BA R BA R A W EST E R N W ASHINGTON STATE COLLEGE
