On the Distribution of Sidon Series

Nakhlé H. Asmar^{∗} and Stephen Montgomery-Smith^{†} University of Missouri–Columbia Columbia, MO 65211 U. S. A.

Abstract

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure μ and dual group Γ. Let E be a Sidon subset of Γ with Sidon constant S(E). Let r_{n} denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all α > 0:

c^{−1}P [∥∑_{n=1}^{N} a_{n}r_{n}∥ ≥ cα] ≤ μ [∥∑_{n=1}^{N} a_{n}γ_{n}∥ ≥ α]

≤ c P [∥∑_{n=1}^{N} a_{n}r_{n}∥ ≥ c^{−1}α]

where a_{1}, …, a_{N} are arbitrary elements of B, and γ_{1} , …, γ_{N} are arbitrary elements of E. We prove a similar result for Sidon subsets of dual objects of compact groups, and apply our results to obtain new lower bounds for the distribution functions of scalar-valued Sidon series. We also note that either one of the above inequalities, even in the scalar case, characterizes Sidon sets.

AMS Subject Classification (1980): 43A46, 43A15, 46E40, 43A77

∗Supported by NSF grant DMS 9102044 †Supported by NSF grant DMS 9001796

1 Introduction

Suppose that G is a compact abelian group with dual group Γ. Denote the normalized Haar measure on G by μ. Let C(G) be the Banach space of continuous complex-valued functions on G. If S ⊂ Γ , a function f ∈ L^{1}(G) is called S −spectral whenever f̂ is supported in S, where here and throughout the paperˆ denotes taking the Fourier transform. The collection of S-spectral functions that belong to a class of functions W will be denoted by W_{S}.

Definition 1.1 A subset E of Γ is called a Sidon set if there is a constant c > 0, depending only on E, such that

∑_{γ∈Γ} |f̂(γ)| ≤ c ∥f∥_{∞}  (1)

for every f ∈ C_{E}(G). The smallest constant c such that (1 ) holds is denoted by S(E) and is called the constant of sidonicity of E, or the Sidon constant of E.

If E = {γ_{j}} ⊂ Γ is a Sidon set and {a_{j}} is a sequence in a Banach space B, then the formal series ∑ a_{j}γ_{j} will be referred to as a B-valued Sidon series. The norm on a given Banach space B will be denoted by ∥ · ∥, or, sometimes, by ∥ · ∥_{B}. It is well-know that Sidon series share many common properties with Rademacher series. The following theorem of Pisier illustrates this fact and will serve as a crucial tool in our proofs.

Theorem 1.2 ([Pi 1, Théorème 2.1]) Suppose that E = {γ_{n}} ⊂ Γ is a Sidon set, that B is a Banach space, and that a_{1}, …, a_{N} ∈ B. There is a constant c_{1}, depending only on the Sidon constant S(E), such that, for every p ∈ [1, ∞[, we have

c_{1}^{−1} (E ∥∑_{n=1}^{N} a_{n}r_{n}∥^{p})^{\frac{1}{p}} ≤ (∫_{G} ∥∑_{n=1}^{N} a_{n}γ_{n}∥^{p} dμ)^{\frac{1}{p}} ≤ c_{1} (E ∥∑_{n=1}^{N} a_{n}r_{n}∥^{p})^{\frac{1}{p}} .

(2)

In view of this similarity between Sidon series and Rademacher series, it is natural to ask how the distribution function of a Sidon series compares to the distribution function of a Rademacher series. Our main result provides an answer to this question.

Theorem 1.3 Suppose that E = {γ_{n}} ⊂ Γ is a Sidon set, and let B denote an arbitrary Banach space. There is a constant c > 0 that depends only on the Sidon constant S(E), such that for all a_{1}, …, a_{N} ∈ B, and all α > 0, we have

c^{−1}P [∥∑_{n=1}^{N} a_{n}r_{n}∥ ≥ cα] ≤ μ [∥∑_{n=1}^{N} a_{n}γ_{n}∥ ≥ α] ≤ cP [∥∑_{n=1}^{N} a_{n}r_{n}∥ ≥ c^{−1}α] .

(3)

Thus the distribution functions of Sidon and Rademacher series are equivalent. Our proof of this result combines well-known properties of Sidon series with Lemma 2.3 below. This Lemma provides sufficient conditions for the equivalence of distribution functions. It applies as well in the setting of noncommutative groups yielding an analogue of Theorem 1.3. Using the estimates of [Mo], we obtain sharp lower bound estimates on the distribution of scalar-valued Sidon series on compact abelian groups.

We should note that the topological implication of our main result is much easier to show, as is done in [Pi 1], that is, the measure topology on the spaces of Rademacher series and the Sidon series are equivalent.

2 A principle for the equivalence of distribution functions

All random variables are defined on some probability space (Ω, M, dP). We denote the set of positive integers by N, the set of integers by Z, and the circle group by T. All other notation is as in Section 1. We start with a couple of preliminary lemmas.

Lemma 2.1 Suppose that f_{1}, f_{2}, …, f_{N} are independent identically distributed random variables , and let f be a function with the same distribution as the f_{j}’s such that

P [|f| ≥ α] ≥ \frac{θ}{N}

where α and θ are positive numbers. Then

P [ sup_{1≤j≤N} |f_{j}| ≥ α] ≥ \frac{θ}{1 + θ}.

Proof. We first show that, for θ > 0 and N ∈ N, we have

(1 − \frac{θ}{N})^{N} ≤ \frac{1}{(1 + θ)}.  (4)

This follows from the inequalities

(1 − \frac{θ}{N})(1 + \frac{θ}{N}) ≤ 1

and

(1 + \frac{θ}{N})^{N} ≥ 1 + θ.

Using (4 ) and independence, we get:

P [sup_{j} |f_{j}| ≥ α] = 1 − P [sup_{j} |f_{j}| < α]
= 1 − ∏_{j} P [|f_{j}| < α]
≥ 1 − (1 − \frac{θ}{N})^{N}
≥ 1 − \frac{1}{(1 + θ)}
= \frac{θ}{(1 + θ)}.

Lemma 2.2 Suppose that f_{1}, f_{2}, …, f_{N} are independent identically distributed random variables , and let f be a function with the same distribution as the f_{j}’s such that

P [|f| ≥ α] ≤ \frac{θ}{N}

where α and θ are positive numbers. Then, we have

P [sup_{j} |f_{j}| ≥ α] ≤ θ.  (5)

Proof. We start with the inequality

(1 − x)^{n} ≥ 1 − nx (0 ≤ x ≤ 1),

which can be easily proved by using induction and the inequality

(1 − x)(1 − y) ≥ 1 − x − y (x, y ≥ 0).

Hence, for all 0 ≤ θ ≤ N, N ∈ N, we have

(1 − \frac{θ}{N})^{N} ≥ 1 − θ (0 ≤ θ ≤ N, N ∈ N).  (6)

The proof of (5) is now straightforward, using (6) and independence:

P [sup_{j} |f_{j}| ≥ α] = 1 − P [sup_{j} |f_{j}| < α]
= 1 − ∏_{j} P [|f_{j}| < α]
= 1 − ∏_{j}(1 − P [|f_{j}| ≥ α])
≤ 1 − (1 − \frac{θ}{N})^{N}
≤ θ.

Before stating our main Lemma, we recall two well-known inequalities. Let X denote a random variable on a probability space (Ω, M, dP), then, for all y > 0, we have:

P [|X| ≥ yE|X|] ≤ y^{−1} (Chebychev′s Inequality);  (7)

and, for 0 < y < 1, we have

P [|X| ≥ yE|X|] ≥ (1 − y)^{2} \frac{E^{2}|X|}{E|X|^{2}} (P aley − Zygmund Inequality).(8)

See [Ka, Ineq. II, p. 8]. The statement of our main Lemma now follows.

Lemma 2.3 Suppose that X and Y are two Banach valued random variables that are not identically zero, and suppose that {X_{n}} and {Y_{n}} are two sequences of independent Banach valued random variables such that X_{n} is identically distributed with X, and Y_{n} is identically distributed with Y. Suppose that there are constants c_{1} and c_{2} such that, for all positive integers N, we have:

c_{1}^{−1} ∥ sup_{1≤j≤N} ∥X_{j}∥∥_{1} ≤ ∥ sup_{1≤j≤N} ∥Y_{j}∥∥_{1} ≤ c_{1} ∥ sup_{1≤j≤N} ∥X_{j}∥∥_{1} ;  (9)

\begin{matrix}∥sup_{1≤j≤N} ∥X_{j}∥∥ & {2 \atop 1} \\ ∥sup_{1≤j≤N} ∥X_{j}∥∥ & {2 \atop 2} \end{matrix} ≥ c_{2};  (10)

and

\begin{matrix}∥sup_{1≤j≤N} ∥Y_{j}∥∥ & {2 \atop 1} \\ ∥sup_{1≤j≤N} ∥Y_{j}∥∥ & {2 \atop 2} \end{matrix} ≥ c_{2}.  (11)

Then there is a constant c, depending only on c_{1} and c_{2}, such that, for all α > 0, we have:

c^{−1}P [∥X∥ ≥ cα] ≤ P [∥Y ∥ ≥ α] ≤ cP [∥X∥ ≥ c^{−1}α] .  (12)

Proof. We start with the second inequality in (12). Given an arbitrary α = α_{1} > 0 with

0 < P [∥Y ∥ ≥ α_{1}] ,  (13)

choose ν to be the smallest positive integer satisfying:

\frac{1}{2ν} ≤ P [∥Y ∥ ≥ α_{1}] ≤ \frac{1}{ν}.  (14)

From Lemma 2.1, it follows that

\frac{1}{3} ≤ P [ sup_{1≤j≤ν} ∥Y_{j}∥ ≥ α_{1}] .  (15)

Chebychev’s Inequality (7), and (15) imply that

\frac{1}{3} ≤ \frac{1}{α_{1}} ∥ sup_{1≤j≤ν} ∥Y_{j}∥∥_{1} .  (16)

From (9) and (16), we have

\frac{1}{3} ≤ \frac{c_{1}}{α_{1}} ∥ sup_{1≤j≤ν} ∥X_{j}∥∥_{1} .  (17)

Hence, for any α_{2} > 0, (17) implies that

\frac{α_{1}}{3α_{2}c_{1}} ≤ \frac{1}{α_{2}} ∥ sup_{1≤j≤ν} ∥X_{j}∥∥_{1} .  (18)

In particular, if α_{2} = \frac{α_{1}}{6c_{1}} , we get from (18)

2 ≤ \frac{1}{α_{2}} ∥ sup_{1≤j≤ν} ∥X_{j}∥∥_{1} .  (19)

Now we go back to the Paley–Zygmund Inequality (8) and apply it to

sup_{1≤j≤ν} ∥X_{j}∥ with y = \frac{α_{2}}{∥sup_{1≤j≤ν} ∥X_{j}∥∥_{1}} .

Taking into account (10) and noticing from (19) that y ≤ \frac{1}{2}, we get

\frac{1}{4}c_{2} ≤ P [ sup_{1≤j≤ν} ∥X_{j}∥ ≥ α_{2}]
= P [ sup_{1≤j≤ν} ∥X_{j}∥ ≥ dα_{1}]  (20)

where d = \frac{1}{6c_{1}} . Lemma 2.2 , (14), and (20) imply that

P [∥X∥ ≥ dα_{1}] ≥ \frac{c_{2}}{4ν}  (21)
≥ \frac{c_{2}}{4} P [∥Y ∥ ≥ α_{1}] .  (22)

Take c^{−1} = min(d, \frac{c_{2}}{4} ), then (22) shows that

P [∥X∥ ≥ c^{−1}α_{1}] ≥ c^{−1}P [∥Y ∥ ≥ α_{1}] .  (23)

Note that (23) holds for all α_{1} for which (13) is true. For all other values of α_{1}, inequality (23) holds trivially. Thus (23) holds for all α_{1} > 0. Now we repeat the proof with X and Y interchanged. From (23) we get:

P [∥Y ∥ ≥ c^{−1}α_{1}] ≥ c^{−1}P [∥X∥ ≥ α_{1}] ,  (24)

for all α_{1} > 0. Combining (23) and (24), we obtain

c^{−1}P [∥X∥ ≥ α_{1}] ≤ P [∥Y ∥ ≥ c^{−1}α_{1}] ≤ cP [∥X∥ ≥ c^{−2}α_{1}]  (25)

for all α_{1} > 0. Equivalently, we have

c^{−1}P [∥X∥ ≥ cα] ≤ P [∥Y ∥ ≥ α] ≤ cP [∥X∥ ≥ c^{−1}α]  (26)

for all α > 0 , which proves (12).

3 The distribution of Banach valued Sidon series

To prove Theorem 1.3, Lemma 2.3 suggests that we consider independent copies of the given Sidon series. The construction of independent copies of a given trigonometric polynomial on a group G is easily done on the product group. The spectra of the resulting polynomials are supported in a subset of the product of the character group. Our first goal in this section is to study properties of this set. To simplify the presentation, we will treat the commutative and noncommutative cases separately. Throughout this section G will denote a compact abelian group with character group Γ and Haar measure μ. Similar meanings are attributed to G_{j}, Γ_{j}, μ_{j}, respectively.

Definition 3.1 Suppose that E_{j} ≠ ∅ is a subset of Γ_{j}, for j = 1, …, n. The n-fold join of the sets E_{j} is a subset of ∏_{j=1}^{n} Γ_{j}, denoted by ⋁_{j=1}^{n} E_{j} , and

defined by:

⋁_{j=1}^{n} E_{j} = {γ = (γ_{1}, …, γ_{n}) ∈ ∏_{j=1}^{n} Γ_{j} : all but one γ_{j} ∈ E_{j} are 0}.

If E ≠ ∅ ⊂ Γ, the n-fold join of E, denoted by ⋁_{j=1}^{n} E, is the set ⋁_{j=1}^{n} E_{j}, where E_{j} = E for all j = 1, …, n.

Thus a generic element γ of ⋁_{j=1}^{n} E_{j} is of the form γ = (0, 0, …, γ_{j}, 0, …, 0) where γ_{j} ∈ E_{j}. When γ is evaluated at x = (x_{1}, …, x_{j}, …, x_{n}) ∈ G^{n} we get:

γ(x) = γ_{j}(x_{j}).

Suppose that S ⊂ Γ. For the sake of our proof of Theorem 1.3, it turns out that it is sufficient to study the n–fold join of the set {1} × S ⊂ Z × Γ. What is needed is the following simple result. A more general result concerning joins of Sidon sets is presented following the proof of Theorem 1.3.

Lemma 3.2 Suppose E ⊂ Γ is a Sidon set. Let T = {1} × E ⊂ Z × Γ, and let n be an arbitrary positive integer. The n-fold join ⋁_{j=1}^{n} T is a Sidon subset of Z^{n} × Γ^{n} with Sidon constant S (⋁_{j=1}^{n} T ) equal to S(E).

Proof. It is clear that S(T) = S(E) and that S (⋁_{j=1}^{n} T )≥ S(E). Let F

be a trigonometric polynomial with spectrum supported in S (⋁_{j=1}^{n} T ). We can write F as:

F = ∑_{j=1}^{n} f_{j}(t_{j}, x_{j})

where

f_{j} = ∑_{l=1}^{k_{j}} a_{jl}e^{it_{j}}χ_{jl}(x_{j}) = e^{it_{j}} ∑_{l=1}^{k_{j}} a_{jl}χ_{jl}(x_{j}).

For each j = 1, …, n, pick x_{j} so that

|∑_{l=1}^{k_{j}} a_{jl}χ_{jl}(x_{j})| = ∥f_{j}∥_{∞},

and then pick t_{j} so that

e^{it_{j}} ∑_{l=1}^{k_{j}} a_{jl}χ_{jl}(x_{j}) = ∥f_{j}∥_{∞}.

We have

f_{j}(t_{j}, x_{j}) = ∥f_{j}∥_{∞},

and so

F (t_{1}, t_{2}, …, t_{n}, x_{1}, x_{2}, …, x_{n}) = ∑_{j=1}^{n} ∥f_{j}∥_{∞} ≥ (S(E))^{−1} ∑_{j=1}^{n} ∑_{l=1}^{k_{j}} |a_{jl}|.

Hence (1) holds for F with c = S(E), and the proof is complete.

We still need one ingredient for the proof of Theorem 1.3. This is the Khintchin–Kahane theorem.

Theorem 3.3 ([Pi 1, Théorème K]) If 0 < p < q < ∞, there is a constant K_{p,q} such that

(E ∥∑_{j=1}^{n} a_{j}r_{j}∥^{q})^{\frac{1}{q}} ≤ K_{p,q} (E ∥∑_{j=1}^{n} a_{j}r_{j}∥^{p})^{\frac{1}{p}}  (27)

for any sequence {a_{j}} in a Banach space B

Proof of Theorem 1.3 Let f(x) = ∑_{n=1}^{N} a_{n}γ_{n}(x) (x ∈ G) be a trigonometric polynomial with γ_{n} ∈ S, and let Y = e^{it} ∑_{n=1}^{N} a_{n}γ_{n}(x) where (t, x) ∈ T × G. Clearly, f and Y have the same distribution functions. We apply Lemma 2.3 with

X = ∑_{n=1}^{N} a_{n}r_{n} and Y = e^{it} ∑_{n=1}^{N} a_{n}γ_{n}(x).

Given ν ∈ N, we construct a sequence of independent random variables {Y_{j}} on T^{ν} × G^{ν}, identically distributed with Y, in the obvious way: for (t, x) = (t_{1}, t_{2}, …, t_{ν}, x_{1}, …, x_{ν}) ∈ T^{ν} × G^{ν}, let

Y_{j}(t, x) = Y (t_{j}, x_{j}) = e^{it_{j}} ∑_{n=1}^{N} a_{n}γ_{n}(x_{j}).  (28)

Write γ_{nj} (n = 1, …, N; j = 1, …, ν) for the character in Γ^{ν} given by:

γ_{nj}(x) = γ_{n}(x_{j})

for all x ∈ G^{ν}. Let ℓ_{∞}^{ν}(B) denote the Banach space consisting of vectors a = (a_{1}, …, a_{ν}), where a_{j} ∈ B, equipped with the norm

∥a∥_{ℓ_{∞}^{ν}(B)} = sup_{1≤j≤ν} ∥a_{j}∥ .

Let a_{nj} ∈ ℓ_{∞}^{ν}(B) be the vector whose components are all zero except the j−th component is equal to a_{n}: a_{nj} = a_{n} (δ_{ij})_{i=1}^{ν} , n = 1, …, N ; j = 1, …, ν. Let

Y(t, x) = ∑_{j=1}^{ν} ∑_{n=1}^{N} a_{nj}e^{it_{j}}γ_{nj}(x) ((t, x) ∈ T^{ν} × G^{ν}).  (29)

We clearly have:

Y(t, x) = (Y_{1}(t, x), Y_{2}(t, x), …, Y_{ν}(t, x))  (30)

for all (t, x) ∈ T^{ν} × G^{ν}. For j = 1, …, ν, let

X_{j}(t) = ∑_{n=1}^{N} a_{n}r_{nj}(t).

Corresponding to Y, construct a Rademacher sum X with values in ℓ_{∞}^{ν}(B):

X (t) = ∑_{j=1}^{ν} ∑_{n=1}^{N} a_{nj}r_{nj}(t) (t ∈ [0, 1]),  (31)

where (r_{nj})_{n=1}^{N}_{j=1}^{ν} is an enumeration of distinct Rademacher functions, and a_{nj} is as above. Note that Y is a Sidon series with spectrum supported in the join ⋁_{j=1}^{ν}{1} × E = ⋁_{j=1}^{n} T of T = {1} × E. Pisier’s Theorem 1.2 implies that

c_{1}^{−1} (E sup_{1≤j≤ν} ∥X_{j}∥) ≤ ∫_{G^{ν}} sup_{1≤j≤ν} ∥Y_{j}∥ ≤ c_{1} (E sup_{1≤j≤ν} ∥X_{j}∥)  (32)

where c_{1} depends only on S (⋁ T), and hence only on S(E), by Lemma 3.2. We have thus obtained (9). It remains to prove (10) and (11). These are consequences of (27) and Pisier’s Theorem 1.2 applied to the random variables X and Y above. Indeed, applying (27) with p = 1 and q = 2, we obtain:

(E ( sup_{1≤j≤ν} ∥X_{j}∥)^{2})^{\frac{1}{2}} ≤ K_{1,2}E sup_{1≤j≤ν} ∥X_{j}∥  (33)

which proves (10). To get (11), we apply Pisier’s Theorem 1.2, with p = 2 to the functions X and Y, then use (32) and (33) again.

Remark 3.4 It is worth noting that either of the inequalities (3) characterizes Sidon sets. For the first inequality, the assertion follows from the definition of Sidon sets. For the second inequality, this follows by Pisier’s characterization of Sidon sets [Pi 3].

In our original proof of Theorem 1.3, we worked directly with the function f = ∑ a_{j}γ_{j} whose spectrum is supported in a Sidon set E. The result that we needed concerned the Sidon constant of the n–fold join of E. We present this result in the next theorem, because of its interest in its own right. As far as we know, the best constant in the theorem below is not known.

Theorem 3.5 Suppose that E is a Sidon subset of Γ and n ≥ 1. Then ⋁_{j=1}^{n} E is a Sidon subset of Γ^{n} with Sidon constant S (⋁_{j=1}^{n} E) ≤ 2πS(E)+1.

In particular, S (⋁_{j=1}^{n} E) is independent of n.

The proof below was kindly communicated to us by D. Ullrich and other people after him. It is an easy consequence of the following lemma which, as D. Ullrich also remarked, may be well-known to probabilists.

Lemma 3.6 For j = 1, …, n, let K_{j} denote a compact topological space. Suppose that f_{j} ∈ C(K_{j}), j = 1, …, n. Suppose further that 0 is in the convex hull of f_{j}(K_{j}), for each j. Define f ∈ C (∏_{j=1}^{n} K_{j}) by f(x) = ∑_{j=1}^{n} f_{j}(x_{j}). Then

π∥f∥_{∞} ≥ ∑_{j=1}^{n} ∥f_{j}∥_{∞}.

Moreover, the constant π is best possible.

Proof. For all θ ∈ [−π, π[, and all x ∈ ∏_{j=1}^{n} K_{j}, we clearly have:

∥f∥_{∞} ≥ ℜ (e^{iθ}f(x)) = ∑_{j=1}^{n} ℜ (e^{iθ}f_{j}(x_{j})) .

Choose b_{j} ∈ K_{j} such that |f_{j}(b_{j})| = ∥f_{j}∥_{∞}. Write |f_{j}(b_{j})| = e^{iψ_{j}}f_{j}(b_{j}) for ψ_{j} ∈ [−π, π[. Now, for each θ ∈ [−π, π[, ℜ (e^{iθ}f_{j}(b_{j})) may, or may not, be

nonnegative. If ℜ (e^{iθ}f_{j}(b_{j})) ≥ 0, set a_{j}(θ) = b_{j}. If ℜ (e^{iθ}f_{j}(b_{j})) < 0, define

a_{j}(θ) ∈ K_{j} to be any element of K_{j} such that ℜ (e^{iθ}f_{j}(a_{j}(θ))) ≥ 0. This is possible since the convex hull of f_{j}(K_{j}) contains 0. Thus, for all θ ∈ [−π, π[, we have:

ℜ (e^{iθ}f_{j}(a_{j}(θ))) ≥ max{0, cos(θ − ψ_{j})∥f_{j}∥_{∞}}.

Hence,

∥f∥_{∞} ≥ ∑_{j=1}^{n} max{0, cos(θ − ψ_{j})∥f_{j}∥_{∞}}.

Now integrating both sides of the last inequality with respect to θ ∈ [−π, π) we obtain

2π∥f∥_{∞} ≥ 2 ∑_{j=1}^{n} ∥f_{j}∥_{∞}

from which the desired inequality follows. To see that we have the best constant in the statement of the Lemma, we consider the following example. For each j = 0, 1, …, 2n − 1, we let K_{j} = \{0, 1\}, say. Define f_{j}(0) = 0 and f_{j}(1) = e^{ij \frac{π}{n}}. Clearly we have ∥f_{j}∥_{∞} = 1. To compute ∥f∥_{∞}, notice that

∥f∥_{∞} = ℜ (e^{iθ} ∑_{j=0}^{2n−1} f_{j}(x_{j}))

for some 0 ≤ θ < 2π and some x_{j} ∈ K_{j}. Thus it is clear that

∥f∥_{∞} = ∑_{j=0}^{n−1} sin(j \frac{π}{n} + θ),

where 0 ≤ θ ≤ \frac{π}{n}. Thus, as n → ∞,

∥f∥_{∞} ≈ n ∫_{0}^{1} sin(πx) dx = \frac{2n}{π} = \frac{1}{π} ∑_{j=0}^{2n−1} ∥f_{j}∥_{∞}.

Corollary 3.7 Suppose E_{j} ⊂ Γ_{j}, S(E_{j}) ≤ κ, for each j = 1, …, N, and 0 ∉ E_{j}. Then S (⋁_{j=1}^{N} E_{j}) ≤ πκ.

Proof. Let E = ⋁_{j=1}^{N} E_{j}, and suppose that f ∈ C_{E}(∏_{j=1}^{N} G_{j}). Then f = ∑_{j=1}^{N} f_{j} with f_{j} ∈ C_{E_{j}}(G_{j}). The fact that 0 ∉ E_{j} shows that f_{j} has mean 0, so Lemma 3.6 may be applied:

π∥f∥_{∞} ≥ ∑ ∥f_{j}∥_{∞} ≥ κ^{−1} ∑_{j} ∑_{γ∈E_{j}} |f̂_{j}(γ)|
= κ^{−1} ∑_{γ∈E} |f̂(γ)|.

To prove Theorem 3.5, apply Corollary 3.7, after removing 0 from E and putting it back, if necessary.

The following is a typical application of Theorem 1.3. It amounts to transferring, via Theorem 1.3, a known result about scalar valued Rademacher series to Sidon series. We need a definition.

Definition 3.8 Suppose that a = (a_{n})_{n=1}^{∞} ∈ ℓ^{2}. Define the sequence (a_{n}^{∗})_{n=1}^{∞} to be the nondecreasing rearrangement of the terms |a_{n}|. If 0 < t < ∞, we define

K_{1,2}(a, t) = ∑_{n=1}^{[t^{2}]} a_{n}^{∗} + t ( ∑_{[t^{2}]+1}^{∞}(a_{n}^{∗})^{2})^{\frac{1}{2}}

where [t] denotes the greatest integer part of t.

The next result follows from Theorem 1.3, and [Mo]. The latter is the following result for Rademacher series.

Theorem 3.9 Suppose that E = (γ_{n}) ⊂ Γ is a Sidon set. There is a constant c > 0 that depends only on S(E) such that for all a = (a_{n})_{n=1}^{∞} ∈ ℓ^{2} we have

μ [|∑_{n=1}^{∞} a_{n}γ_{n}| ≥ cK_{1,2}(a, t)] ≤ ce^{−c^{−1}t^{2}}

and

μ [|∑_{n=1}^{∞} a_{n}γ_{n}| ≥ c^{−1}K_{1,2}(a, t)] ≥ c^{−1}e^{−ct^{2}}

for all t > 0.

Thus it is possible to calculate rearrangement invariant norms of scalar valued Sidon series in the spirit of Rodin and Semyonov [RS].

We note that these results generalize to noncommutative compact groups with no further difficulties. We follow the notation of [Pi 2], Section 5: G is a compact group; Σ is the dual object of G; μ is the normalized Haar measure on G. Thus Σ is the set of equivalence classes of the irreducible representations of G. For each ι ∈ Σ, we let U_{ι} denote a representing element of the equivalence class. Thus U_{ι}(x) is, for each x ∈ G, a unitary operator on a fixed finite dimensional Hilbert space H_{ι}. The dimension of H_{ι} will be denoted by d_{ι}. For further details, we refer the reader to [Pi 2], and [HR].

In analogy with Definition 1.1, a subset S ⊂ Σ is called a Sidon set if there is a constant c such that

∑ d_{ι}tr |f̂(ι)| ≤ c∥f∥_{∞}

for every f ∈ C_{S}(G), where C_{S}(G) is defined as in the abelian case.

Let I be a countable indexing set. An analogue of the Rademacher functions is defined as a sequence, {ε_{ι}}_{ι∈I}, of independent random variables, each ε_{ι} being a random d_{ι} × d_{ι} orthogonal matrix, uniformly distributed on the orthogonal group O(d_{ι}). These functions are studied in [Pi 2] and [MP]. Note that we have the analogue of the Khintchin–Kahane inequality due to Pisier–Marcus, [MP, Corollary 2.12, p. 91].

The analogue of Pisier’s Theorem 1.2 can be easily established in this setting by repeating the proof in [Pi 1] and making use of the properties of Sidon sets on noncommutative groups. All of these properties are found in [HR, Theorem (37.2)]. For ease of reference, we state the result below, and omit the proof.

Theorem 3.10 Suppose that P = (ι) ⊂ Σ is a Sidon set, that B is a Banach space, and that M_{ι} is a d_{ι} × d_{ι} matrix with values in B. Then there is a constant c_{1}, depending only on the Sidon constant S(P), such that, for every p ∈ [1, ∞[, we have

c_{1}^{−1} (E ∥∑_{ι∈F} d_{ι}tr (ε_{ι}M_{ι})∥^{p})^{\frac{1}{p}} ≤ (∫_{G} ∥∑_{ι∈F} d_{ι}tr (U_{ι}M_{ι})∥^{p})^{\frac{1}{p}}

≤ c_{1} (E ∥∑_{ι∈F} d_{ι}tr (ε_{ι}M_{ι})∥^{p})^{\frac{1}{p}} (34)

for any finite subset F of P

The result concerning the join of Sidon sets in duals of compact groups is a straight analogue of Theorem 3.5. We omit even the statement. We have thus all the necessary ingredients to prove a noncommutative version of our main Theorem 1.3.

Theorem 3.11 Let G be a compact group with dual object Σ. Suppose that P = {ι} ⊂ Σ is a Sidon set. For n = 1, …, N, let a^{n} denote a d_{ι_{n}} × d_{ι_{n}} matrix with entries in a Banach space B. There is a constant c > 0 such that, for all α > 0, we have:

c^{−1}P [∥∑_{n=1}^{N} d_{ι_{n}}tr (ε_{ι_{n}}a^{n})∥ ≥ cα] ≤ μ [∥∑_{n=1}^{N} d_{ι_{n}}tr (U_{ι_{n}}a^{n})∥ ≥ α]

≤ cP [∥∑_{n=1}^{N} d_{ι_{n}}tr (ε_{ι_{n}}a^{n})∥ ≥ c^{−1}α] .(35)

We close our paper by pointing out to the interested reader that, under suitable conditions, these methods also apply to commensurate sets of characters of Pelczyński [Pe], as indeed they do also to topological Sidon sets, also described by Pelczyński loc. sit.

References

[HR] E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis”, Vol. 2, 152, Springer–Verlag, Berlin and New York, 1970.

[Ka] J.P. Kahane,“Some Random Series of Functions”, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge 1985.

[MP] M. Marcus and G. Pisier, “Random Fourier Series with Applications to Harmonic Analysis”, Annals of Mathematics Studies, Princeton University Press and University of Tokyo Press, Princeton, 1981.

[Mo] S. Montgomery-Smith, The distribution of Rademacher sums, Proc. AMS, 109, (1990), 517–522.

[Pe] A. Pelczyński, Commensurate sequences of characters, Proc. Amer. Math. Soc. 104, (1988), 525–531.

[Pi 1] G. Pisier, Les inégalités de Kahane–Khintchin d’après C. Borell, Séminaire sur la géometrie des éspaces de Banach, École Polytechnique, Palaiseau, Exposé No. VII, (1977–1978).

[Pi 2] G. Pisier, De nouvelles caractérisations des ensembles de Sidon, Mathematical Analysis and its Applications, Part B, Edited by L. Nachbin, Advances in Mathematics Supplementary Studies, 7B, (1981), 685–726.

[Pi 3] G. Pisier, Ensembles de Sidon et processus gaussiens, C. R. Acad. Sc. (Paris), Série A, 286, (1978), 671–674.

[RS] V.A. Rodin and E.M. Semyonov, Rademacher series in symmetric spaces, Analyse Math. 1, (1975), 207–222.