C O L L O Q U I U M M A T H E M A T I C U M VOL. 83 2000 NO. 2

VOL. 83 2000 NO. 2

INTERPOLATION SETS FOR FR ´ ECHET MEASURES

BY

J. C A G G I A N O (STATE UNIVERSITY, AR)

Abstract. We introduce various classes of interpolation sets for Fr´echet measures—

the measure-theoretic analogues of bounded multilinear forms on products of C(K) spaces.

1. Introduction. The classical theory of interpolation sets in a harmo- nic-analytic context can be roughly described as the study of norm prop- erties of “one-dimensional” objects (bounded linear forms) in relation to some underlying spectral set. The study of interpolation sets for naturally multi-dimensional structures has developed only in the last twenty years;

see [GMc], [GS2]. In this work, it is our aim to examine certain harmonic- analytic interpolation properties of Fourier transforms of Fr´echet measures—

the measure-theoretic counterparts of multi-linear forms on products of C

(K) spaces. There are some interesting departures from the one-dimen- sional theory.

Definition 1 ([B5, Def. 1.1]). Let X

, . . . , X

be locally compact spaces with respective Borel fields A

, . . . , A

. A set function µ : A

×. . .×A

→ C is an F

-measure if, when n − 1 coordinates are fixed, µ is a measure in the remaining coordinate. When the measure spaces are arbitrary or understood, we denote the space of F

-measures by F

= F

( A

, . . . , A

).

For our purposes, each space X

will be the circle group T. There is a natural identification between the space of F

-measures on T × . . . × T and the space of bounded n-linear forms on C(T) × . . . × C(T) [B4, Thm. 4.12].

Denoting this identification by

β

↔ η,

we define the Fourier transform of an F

-measure η on T

to be the function

2000 Mathematics Subject Classification: 43A25, 43A46, 46A32.

This is a portion of my doctoral thesis. I thank my advisor, Ron Blei, for his patience and encouragement. I also thank the Department of Mathematics at the University of Connecticut for its generous support during my graduate studies.

[161]

on Z

given by b

η(m

, . . . , m

) = β

(e

, . . . , e

)

=

\

Tⁿ

e

⊗ . . . ⊗ e

η(dt

, . . . , dt

)

=

\

Tⁿ⁻¹

e

⊗ . . . ⊗ e

\

T

e

η(dt

, . . . , dt

).

The integral above is defined iteratively, i.e.,

\

T

e

η(dt

, . . . , dt

) ∈ F

(T, . . . , T);

see [B4, Lemma 4.9] for details.

The space of F

-measures on X ×Y (referred to as the space of bimeasures on X × Y in the literature) is a convolution Banach *-algebra [GS1] whose structure extends that of the space of measures on X × Y. Convolution of F

-measures is not well defined in general when n > 2 [GS3], essentially because there is no general Grothendieck-type inequality for n > 2. If we restrict our attention to the so-called projectively bounded Fr´echet measures, we have a well defined convolution, as well as suitable extensions of the Grothendieck inequality. The class of completely bounded multi-linear forms has also been considered as a natural class of F

-measures which satisfies a Grothendieck-type inequality; see [CS], [ZS], [Y].

Definition 2 ([B5]). Let µ ∈ F

(T, . . . , T), and let E

, . . . , E

be finite subsets of the unit ball of L

(T). For (f

, . . . , f

) ∈ E

× . . . × E

define (1) φ

(f

, . . . , f

) =

\

Tⁿ

f

⊗ . . . ⊗ f

µ(dt

, . . . , dt

).

Let

(2) kµk

= sup {kφ

k

:

E

⊂ Ball(L

(T)), |E

| < ∞, j = 1, . . . , n}.

Then µ is projectively bounded if kµk

< ∞. The space of projectively bounded F

-measures on T ×. . .×T is denoted by PBF

= PBF

(T, . . . , T).

The class of projectively bounded F

-measures is a non-empty proper

subspace of F

for n > 2, and PBF

= F

for n < 3 (see [B5]). Pro-

jectively bounded F

-measures obey a Grothendieck-type inequality in the

sense that b µ ∈ e V

(Z, . . . , Z) for all µ ∈ PBF

. To see this, let E

=

{e

, . . . , 1, . . . , e

}, and let m

, . . . , m

∈ [N] = {−N, . . . , −1,

0, 1, . . . , N }. Then

(3) kb µ1

k

= kφ

k

≤ kµk

. Since b µ1

→ b µ pointwise, we see immediately that b µ ∈ e V

(Z, . . . , Z).

Given E ⊂ Z

and m ≤ n, we define

B

(E) = {φ ∈ ℓ

(E) : ∃µ ∈ F

, b µ(j

, . . . , j

) = φ(j

, . . . , j

) on E }, with

kφk

= inf {kµk

: b µ = φ on E }, and

P B

(E) = {φ ∈ ℓ

(E) : ∃µ ∈ PBF

, b µ(j

, . . . , j

) = φ(j

, . . . , j

) on E }, with

kφk

= inf {kµk

: b µ = φ on E }.

A word about the condition m ≤ n: there are certain canonical contain- ments in F

(T, . . . , T), which yield corresponding containments in the re- striction algebras defined above. Consider the case n = 3. We have F

(T

) F

(T

, T) F

(T, T, T), so B

(Z

) B

(Z

) B

(Z

). For certain subsets of Z

we may have equality of restriction algebras; see Def. 11.

For a given Banach space A of functions on Z

and S ⊂ Z

, we use the notation [A] |

to denote the quotient space A/J

, where

J

= {f ∈ A : f = 0 on S}.

Similarly, for a given Banach space B of functions on T

and S ⊂ Z

, we use the notation [B]

to denote {f ∈ B : b f = 0 on S

}. We define V

= V

(T, . . . , T) ≡ ˆ N

k=1

C(T). The Banach space dual of V

is F

= F

(T, . . . , T) [B4, Thm. 4.12]. The Banach space dual of F

(Z, . . . , Z) is the space e V

(Z, . . . , Z), given by

V e

(Z, . . . , Z) = {φ ∈ ℓ

(Z

) : φ = lim

k

φ

pointwise, sup

k

kφ

k

< ∞}, where k k

denotes the norm in V

(Z, . . . , Z). We will use k · k

to denote the injective tensor norm, and we note that it is straightforward to show that F

(Z, . . . , Z) is canonically isomorphic to ˇ N

j=1

ℓ

(Z).

2. Interpolation sets 2.1. PBF

-Sidon sets

Proposition 3 ([GS2, Thm. 1]). b η ∈ e V

(Z, Z) for all η ∈ F

(T, T).

P r o o f. Choose Grothendieck probability measures ν

, ν

([G, Corol- laire 2, p. 61], [GS1, Thm. 1.2]), so that η extends to L

(T, ν

) × L

(T, ν

).

Still denoting this extension by η, we have

kηk ≤ K

kηk

.

Let S : L

(T, ν

) → L

(T, ν

) satisfy η(f, g) = hSf, gi and let {a

} ∈ F

(Z, Z). Finally, choose finite subsets A and B of Z. Then

 X

j∈A, k∈B

a

η(j, k) b    = K

kηk

X

j∈A, k∈B

a

 Se

K

kηk

, e

≤ 4K

kηk

k{a

}k

.

The last inequality follows immediately from the Grothendieck inequality [LP, Thm. 2.1].

We note that Proposition 3 is equivalent to ℓ

(Z) ˇ ⊗ ℓ

(Z) ⊂ C(T) ˆ ⊗C(T) under the correspondence

{a

} ↔ X

m,n

a

e

e

.

Definition 4. A set S ⊂ Z × Z is called PBF

-Sidon if P B

(S) = [e V

(Z, . . . , Z)] |

. The PBF

-Sidon constant of S is

γ

= sup {kφk

: kφk

= 1 }.

In [GS1] (resp. [GS2]), the authors define BM-Sidon (resp. BM-inter- polation) sets to be those subsets E of b G × b H for which P B

(E) = C(E), where G and H are LCA groups. The case n = 2 in Definition 4 is different, and we see that BM-Sidon sets are necessarily PBF

-Sidon.

The sections of PBF

-Sidon sets behave as expected; let E be PBF

- Sidon, and let S ⊂ {1, . . . , n} be an ordered subset with |S| = m. Define the projection π

: Z

→ Z

in the obvious way. Then the (n − m)-section

E ∩ π

(j

, . . . , j

)

is PBF

-Sidon. To show this, we need only interpolate elementary ten- sors in e V

(Z, . . . , Z). Any such tensor ψ is extendible to a tensor ψ ∈ V e

(Z, . . . , Z) in the obvious way. Since E is PBF

-Sidon, we can find a pro- jectively bounded Fr´echet measure µ

which interpolates ψ on E. Viewing µ

as an n-linear form, we see that we obtain a bounded (n −m)-linear form by simply fixing the coordinates of µ

corresponding to S. This restriction is projectively bounded, and interpolates the original tensor ψ on E ∩ π

. For S ⊂ Z × Z we let I

(T, T) = {f ∈ [V

(T, T)]

: b f ∈ [ℓ

⊗ ℓ ˇ

]

}. The proof of the following theorem is straightforward.

Theorem 5. Let S ⊂ Z × Z. The following are equivalent:

(i) I

(T, T) = [ V

(T, T)]

(i.e., S is PBF

-Sidon).

(ii) ∃C > 0 with k b f k

≤ Ckfk

, ∀f ∈ I

(T, T).

(iii) I

(T, T) = [L

(T) ˆ ⊗ L

(T)]

.

Proposition 6. (i) Let E, F ⊂ Z be Sidon. Then E ×F is PBF

-Sidon.

(ii) Let f : Z → Z be strictly monotone. Then the graph of f is PBF

- Sidon.

P r o o f. (i) Clearly [e V

(Z, Z)] |

= e V

(E, F ). We show e V

(E, F ) ⊂ P B

(E × F ). By a standard compactness argument, we need only interpo- late restrictions of elementary tensors to E × F. Let α

, α

be the Sidon constants of E and F , and let φ ⊗ ψ be an elementary tensor of norm 1 in V

(E, F ). Then we can find measures µ and ν such that b µ(m)b ν(n) = φ(m)ψ(n) for any (m, n) ∈ E × F, with kµ ⊗ νk

≤ α

α

.

(ii) As shown in [GS1, Thm. 6.3], any bounded sequence on graph f can be interpolated by the transform of an F

-measure.

The existence of other examples of PBF

-Sidon sets is not known. In one dimension, the use of Riesz products as interpolating measures suggests a number of arithmetic criteria on subsets as sufficient conditions for satisfac- tion of the Sidon property. There is no clear connection between arithmetic properties of a given subset of Z

and the PBF

-Sidon property, making the question of sufficiency somewhat more delicate.

As in the one-dimensional case, there is an approximate interpolation condition for PBF

-Sidon sets.

Proposition 7. Let E ⊂Z×Z. If there are 0<δ <1 and 0<C <∞ such that for all f, g ∈ Ball

(ℓ

(Z)), there is µ ∈ F

(T, T), kµk

< C, satisfying (4) kf ⊗ g − b µ k

< δ,

then E is PBF

-Sidon.

P r o o f. We show that [C(T) ˆ ⊗ C(T)]

⊂ [ℓ

⊗ ℓ ˇ

]

. Choose a polyno- mial f with support in E, and select ω

, ω

∈ {−1, 1}

. The canonical pro- jections r

: {−1, 1}

→ {−1, 1} given by r

(ω) = ω(j) are the Rademacher functions. Choose an F

-measure µ

satisfying

kb µ

(j, k) − r

(ω

)r

(ω

) k

< δ.

By (4) and duality,  

 X

(j,k)∈E

f (j, k)r b

(ω

)r

(ω

)    ≤  

 X

(j,k)∈E

f (j, k)(r b

(ω

)r

(ω

) − b µ

(j, k))   

+    X

(j,k)∈E

f (j, k)b b µ

(j, k)   

≤ δk b f k

+ kfk

kµ

k

. Taking suprema over all choices of ω

and ω

, we obtain

k b f k

≤ 4C

1 − δ kfk

.

The factor 4 appears due to the consideration of real and imaginary parts in the calculation of the injective norm of b f .

We note that it is straightforward to show that the union of PBF

-Sidon sets of the type described in Proposition 6 is again PBF

-Sidon, whereas the union problem in general remains open. An analogous approximate in- terpolation condition holds for PBF

-Sidon sets.

That the diagonal of Z × Z is PBF

-Sidon allows us to demonstrate a fundamental difference between multiplier properties of F

-measures and F

-measures. Define U : F

(T, T) × F

(T, T) → F

(T, T) by U (µ, ν) = µ ∗ν.

Then U is a bounded bilinear operator [GS1, Thm. 2.6]. Interestingly, U is not bounded on L

(T

) × F

(T, T), in direct contrast with the situation for ( F

-)measures.

Proposition 8. Let 2 < p < ∞. Then U is not a bounded operator on L

(T

) × F

(T, T).

Let ∆ denote the diagonal in Z × Z, and let f be a ∆-polynomial, f (s, t) = X

a

e

. For ω ∈ {−1, 1}

, let

f

(s, t) = X

a

e

r

(ω).

Since ∆ is PBF

-Sidon, we can find µ

∈ F

(T, T) such that b

µ(j, j) = r

(ω), j ∈ Z,

and kµ

k

≤ γ

for all ω. Suppose kUk = C < ∞. Then f = f

∗ µ

= U (f

, µ

)

and kfk

≤ Cγ

kf

k

, which implies that kfk

≤ Cγ

E

kf

k

. Applying Khinchin’s inequalites, we obtain

kfk

≤ C

γ

E

\

T

\

T

   X

a

e

r

(ω)   

p

= C

γ

\

T

\

T

E

   X

a

e

r

(ω)   

p

≤ C

γ

p

 X

|a

|



, which shows that ∆ is a Λ(p)-set, a contradiction.

We now consider PBF

-Sidon sets for n > 2. The space of F

-measures on T

is denoted by M(T

).

Lemma 9. M(T

) ⊂ PBF

(T, . . . , T).

P r o o f. Let µ ∈ M(T

), and choose finite subsets E

, . . . , E

in the unit ball of L

(T). Then

kφ

k

= sup

kβk_Fn≤1

 X

f1∈E1,...,fn∈En

β(f

, . . . , f

)

\

Tⁿ

(f

⊗ . . . ⊗ f

) µ(dt × . . . × dt)   

≤ sup

kβk_Fn≤1

\

Tⁿ

 X

f1∈E1,...,fn∈En

β(f

, . . . , f

)(f

⊗ . . . ⊗ f

)    |µ|(dt × . . . × dt)

≤ 2

kµk

.

Corollary 10. If E

, . . . , E

are Sidon, then E

× . . . × E

is PBF

- Sidon.

P r o o f. We need only recall that B(E

× . . . × E

) = e V

(E

, . . . , E

).

2.2. F

/ F

-sets

Definition 11. Let m > n ≥ 0. For n > 0, a set E ⊂ Z

is an F

/ F

- set if B

(E) = B

(E); E ⊂ Z

is an F

/ F

-set if B

(E) = ℓ

(E).

We define PBF

/ PBF

, PBF

/ F

, and F

/ PBF

sets analogously.

In this terminology, Sidon sets are F

/ F

-sets and BM-Sidon sets are F

/ F

- sets. In [GS2], the authors use the term BM/B-sets for those subsets of the dual of an LCA group whose bimeasure restriction algebra coincides with the measure restriction algebra. In our terminology, these are F

/ F

-sets.

We see immediately that any Sidon set in Z

is F

/ F

, F

/ F

(and hence F

/ F

for any n < m), PBF

/ PBF

and F

/ PBF

.

The proof of Proposition 6(i) shows that we need not step outside the space of measures to interpolate all of e V

(E, F ) when E and F are Sidon.

Thus, we have

Corollary 12. If E and F are Sidon subsets of Z, then E × F is F

/ F

.

There is a partial converse to the previous corollary: if A × A is F

/ F

then A is Sidon. To see this, let ∆

= {(a

, a

) : a

∈ A}. Since the F

/ F

property is inherited by subsets, ∆

is F

/ F

. We claim that

∆

is Sidon in Z × Z. Let φ ∈ ℓ

(∆

). For f, g ∈ C(T), define η

(f, g) = X

j

φ(j) b f (a

)bg(a

).

Then η

is a bounded linear form on V

(T, T) satisfying b

η

(a

, a

) = φ(j).

But ∆

is F

/ F

, and so we can find a measure µ

satisfying b

µ

(a

, a

) = φ(j).

So ∆

is Sidon. Now, let f be an A-polynomial, f (s) = P

j

c

e

, and consider

F (s, t) = X

j

c

e

. If B denotes the Sidon constant of ∆

, we have

k b f k

= X

j

|c

| ≤ B sup

s,t

   X

j

c

e

   = Bkf k

,

and A is Sidon. We also note that the result above need not hold when the factors forming the Cartesian product in Z × Z are different. For example, Z × {n} is F

/ F

. To see this, choose µ ∈ F

(T, T), and let η

be the corresponding bilinear form on C(T) × C(T). Then

b

µ |

= (η

( ·, n) ⊗ e

dt) |

.

This leads to a question. Given two (different) infinite subsets A and B such that A × B is F

/ F

, must A or B be Sidon? We do not know the answer.

Which sets are both F

/ F

and PBF

-Sidon? It is obvious that any Sidon subset of Z × Z is necessarily F

/ F

and PBF

-Sidon. We can glean a bit more. It is straightforward to show

Proposition 13. If S is F

/ F

and PBF

-Sidon then S is Λ(p) for all p < ∞.

We can separate the various interpolation sets described thus far. Let us consider the two-dimensional case. A product of two Sidon sets is PBF

- Sidon and F

/ F

, but not F

/ F

. Z × {n} is F

/ F

but not PBF

-Sidon or F

/ F

, while the diagonal ∆ = {(n, n) : n ∈ Z} is F

/ F

(hence PBF

- Sidon) but not F

/ F

.

In [GS2], the authors ask: if E, F, and G are infinite subsets of Z such that E ∪ F and G are lacunary (E ∩ F = ∅), must (E + F ) × G be an F

/ F

-set? This question turns out to be a “cusp” case, as Theorem 15 and the next lemma demonstrate.

Lemma 14. Let E, F , G and H be infinite subsets of Z with E ∩ F = G ∩ H = ∅, and E ∪ F, G ∪ H lacunary. Choose a one-to-one correspon- dence between N and N

, and enumerate E, F , G and H according to this correspondence:

(8) E = {λ

}, F = {ν

}, G = {̺

}, H = {κ

}, a, b, c ∈ N.

Then U ⊂ (E + F ) × (G + H) given by

U = {(λ

+ ν

) × (̺

+ κ

) }

is not Sidon.

P r o o f. As described in [B3], the scheme above gives rise to a 4/3- product, with Sidon index 8/7.

Theorem 15. If E, F , G, and H are as above, then (E + F ) × (G + H) is not F

/ F

.

P r o o f. We can find a bounded function φ on U which is not the trans- form of a measure on T

. Then φ is a function of twelve variables, but by the linkages described in the lemma we consider φ as a function of a, b, c, and d. Let β

be the bilinear form on C(T) × C(T) given by

β

(f, g) = X

a,b,c,d

φ(a, b, c, d) b f (λ

+ ν

)bg(̺

+ κ

).

An application of the Cauchy–Schwarz inequality gives boundedness of β

, and we easily verify that

β b

(λ

+ ν

, ̺

+ κ

) = φ(a, b, c, d).

Thus U is not F

/ F

, and any set containing U cannot be F

/ F

.

Let A, B ⊂ Z, A ∩ B = ∅, card(A) = card(B) = ∞. For m, n ≥ 2, let E

, . . . , E

be pairwise disjoint infinite subsets of A and let F

, . . . , F

be pairwise disjoint infinite subsets of B. By considering translates of a set of the form (E + F ) × (G + H), we see that (E

+ . . . + E

) × (F

+ . . . + F

) is not an F

/ F

-set.

We can illustrate something of the “tightness” of the original question in [GS2] as it relates to a generalization of an inequality of Littlewood. One avenue of attack on the problem is as follows. Let

E = {λ

}, F = {ν

}, G = {̺

}.

Any element of ℓ

(N

) ˇ ⊗ℓ

(N) naturally induces a bounded bilinear form on C(T) × C(T). For a = {a

} ∈ ℓ

(N

) ˇ ⊗ ℓ

(N), define such a form β

by

β

(f, g) = X

j,k,l

a

f (λ b

+ ν

)bg(̺

).

The problem is solved if we can produce a tensor as above which simultane- ously is not the transform of a measure restricted to (E + F ) × G. But this cannot be done. Littlewood’s mixed-norm inequality in three dimensions [D]

states that if {a

} is any finitely supported tensor, then ka

k

≤ 2 √

2 sup

l

sX

(j,k)

|a

|

,

which implies that ℓ

(N

) ˇ ⊗ ℓ

(N) ⊂ e V

(N, N, N). Since B((E + F ) × G)

contains all elements of e V

(N, N, N), we see that it is impossible to find a

tensor with the desired properties. As a final comment along this line we remark that in [GS2] the authors prove the following:

Proposition 16. Let H be an infinite subgroup of the discrete group Γ , and let K be any infinite subset of Γ . Then H × K is not F

/ F

.

Notice that this is a “limiting case” of (E

+ . . . + E

) × K.

Certain of the “fractional Cartesian products” [B3], [B4] provide exam- ples of PBF

-Sidon sets, PBF

/ F

-sets, and F

/ F

-sets. For completeness, we include some of the ideas of [B3] and [B4]. Let E be a lacunary subset of Z. Let [m] = {1, . . . , m}. Given S ⊂ [m], π

denotes the projection from E

to E

( |S| = card(S)) given by

π

(e

, . . . , e

) = (e

: j ∈ S),

with the |S|-tuple on the right of the equality above ordered canonically.

Let S = {S

: k = 1, . . . , n } be a collection of subsets of [m] whose union is [m]. Further, we require that each element of [m] appears in at least two elements of S. For each k = 1, . . . , n, consider ℓ

(Z

). Let φ ∈ ℓ

(Z

), and for (x

, . . . , x

) ∈ ℓ

(Z

) × . . . × ℓ

(Z

) define

(6) η

(x

, . . . , x

)

= X

~a∈Z^m

φ(~a)x

(π

(~a)) . . . x

(π

(~a)), x

∈ ℓ

(Z

).

In [B1], Blei shows that for all bounded arrays φ, η

is a well defined n- linear form whose norm is bounded by kφk

. As such, η

can be regarded as an n-linear form on C(T

) × . . . × C(T

), or (equivalently) as an F

- measure on the product of the respective Borel fields of the given products of T. Let

V

(Z

) = n φ(~a) =

X

α

ψ

(π

(~a)) . . . ψ

(π

(~a)), ψ

∈ c

(Z

), X

|α

| < ∞ o . Identifying arrays which are the same pointwise on Z

, we obtain a quotient space, with norm

kφk

= inf n X

|β

| : φ(~a) = X

β

ψ

(π

(~a)) . . . ψ

(π

(~a))

pointwise on Z

o

.

V e

(Z

) is the space of arrays on Z

obtained by taking pointwise limits of

uniformly bounded sequences of elements in V

(Z

).

We now transfer the constructions above to F

(T, . . . , T). Let E ⊂ Z be lacunary, and let S = {S

: k = 1, . . . , n } be a cover of [m] with the properties described above. Consider an m-fold enumeration of E : E = {e

: a

∈ N} along with |S

|-fold enumerations of E : E

= {e

}.

Then we define a subset E

of E

by E

= {(e

, e

S2(j1,...,jm)

, . . . , e

Sn(j1,...,jm)

) : e

Si(j1,...,jm)

∈E

∀i}.

We view η

as an F

-measure in the natural way. It is known [B4] that V e

(Z

) can be realized as a restriction algebra of Fourier–Stieltjes trans- forms of measures on T

, namely,

V e

(Z

) = B(E

) = M(T

)/ {µ ∈ M(T

) : b µ = 0 on (E

)

}.

Theorem 17 ([B1]). The n-linear form η

defined by (6) is projectively bounded if and only if φ ∈ e V

(Z

).

Let e

be the combinatorial dimension of E

([BS]). By [B5, Cor. 7.4]

we see that if e

= 1, then E

is PBF

/ F

. This is a generalization of the

“monotone graphs” of Proposition 6.

Theorem 18. Let E be lacunary, and let S be a cover of [m] so that every element of [m] appears in at least two elements of S. If e

> 1, then E

⊂ Z

is PBF

-Sidon, PBF

/ F

, and F

/ F

, but not F

/ PBF

.

P r o o f. E

is PBF

-Sidon and PBF

/ F

since V e

= B(E

) ⊂ P B

(E

) ⊂ e V

|

= e V

.

The last equality follows from the fact that 1

∈ e V

. Next, E

is F

/ F

since (6) is bounded for all arrays ~a. Finally, because (6) can be projectively unbounded for some choice of φ ([B5, Cor. 7.4]) we see that E

is not F

/ PBF

.

I am grateful to the referee of this paper for a number of useful comments and corrections, and for the suggested definition of F

/ F

-sets.

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Department of Computer Science and Mathematics Box 70, Arkansas State University

State University, AR 72467, U.S.A.

Received 6 November 1998; (3653)

revised 6 April 1999