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C O L L O Q U I U M M A T H E M A T I C U M

VOL. LXVIII 1995 FASC. 1

ADDENDUM TO

“ON HILBERT SETS AND

C

Λ

(G)-SPACES WITH NO SUBSPACE ISOMORPHIC TO c

0

BY

DANIEL L I (ORSAY)

Recently, K. Hare proved ([2], Th. 1), in a more general statement, the following: Every subset Λ in the dual Γ of a compact connected abelian group G which does not contain parallelepipeds of arbitrarily large dimen- sion is strictly-2-associated with every non-empty open subset of G. For Γ = Z, we have actually:

Theorem. If Λ ⊆ Z does not contain a Hilbert set , then it is strictly- 2-associated with every non-empty open subset of Z.

P r o o f. It suffices to apply [1], Prop. 4.26, [4], Th. 7, and [3], Th. 4.

Hence K. Hare’s result is true not only for Λ ∈ S

0

, but for every Λ ∈ S.

REFERENCES

[1] J. B o u r g a i n, New Classes of L

p

-Spaces, Lecture Notes in Math. 889, Springer, 1983.

[2] K. E. H a r e, The support of a function with thin spectrum, Colloq. Math. 67 (1994), 147–154.

[3] J.-P. K a h a n e, Sur les fonctions moyenne-p´ eriodiques born´ ees, Ann. Inst. Fourier (Grenoble) 7 (1957), 293–314.

[4] D. L i, On Hilbert sets and C

Λ

(G)-spaces with no subspace isomorphic to c

0

, this volume, 67–77.

UNIVERSIT ´E DE PARIS-SUD MATH ´EMATIQUE

B ˆAT. 425

91405 ORSAY CEDEX, FRANCE

Re¸ cu par la R´ edaction le 15.11.1994

1991 Mathematics Subject Classification: Primary 46B43.

[79]

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