Summability of large powers of logarithm of classic lacunary series and its simplest consequences.

This paper can be considered as a small appendix to my large article [1] in “Algebra & Analysis”. I’m going to discuss here the last result on lacunary series absolute value distribution obtained there, namely, the following

\underline{Theorem 1.}

Let ε > 0 be arbitrarily small.

Let Λ = {m_{k}}_{k=−∞}^{∞} ⊂ ℤ satisfy the Zygmund lacunarity condition, i.e.

sup_{r≠0} card {(k′, k′′) : m_{k′} − m_{k′′} = r} =^{def}R(Λ) < +∞. (∗)

Then for any function fϵL^{2}(𝕋) spectred at Λ and for each subset E ⊂ 𝕋 of positive measure the inequality

∥f∥_{L^{2}(𝕋)}^{2} ≤ exp \{\frac{B(ε, R(Λ))}{μ(E)^{2+ε}}\} ∫_{E} |f|^{2}dμ

holds with some constant B(ε, R(Λ)) depending on ε and R(Λ) only. As it has already been mentioned in [1], the power 2 + ε in the denominator in the exponent cannot be replaced by 1−ε without additional assumptions on Λ even if we restrict ourselves to arcs E instead of arbitrary measurable sets. Nevertheless, it was commonly believed that much stronger inequality must hold for the most classical example of a spectrum Λ, satisfying (∗): that with Hadamard gaps. In particular, it was conjectured that any (arbitrarily large) power of log|f| is integrable while the theorem 1 implies the summability of powers less than \frac{1}{2} only. The following modification of theorem 1 allows to prove this conjecture.

\underline{Theorem 2.}

Under the conditions of theorem 1, suppose in addition that the difference set DΛ =^{def} m_{k′} − m_{k′′} : k′ ≠ k′′} is a Λ(p) - system with p > 2 (recall that a set U ⊂ ℤ is called a Λ(p) - system if every function g ϵ L^{2}(𝕋) spectred at U belongs to L^{p}(𝕋) as well and ∥g∥_{L^{p}(𝕋)} ≤ C∥g∥_{L^{2}(𝕋)} with some constant c > 0 not depending on g).

Then for any function f ϵ L^{2}(𝕋) spectred at Λ and for each subset E ⊂ 𝕋 of positive measure the inequality

∥f∥_{L^{2}(𝕋)}^{2} ≤ exp \{\frac{B}{μ(E)^{\frac{4}{p} +ε}} \} ∫_{E} |f|^{2}dμ

holds with some constant B = B(ε, Λ) > 0 not depending on f and E.

\underline{Proof}:

The following proposition was the crucial point in the proof of the theorem 1.

\underline{Main lemma}:

Let Λ ⊂ ℤ satisfy the Sigmund lacunarity condition, E ⊂ 𝕋, μ(E) ≤ \frac{1}{2}, fϵ L^{2}(𝕋) and spec f ⊂ Λ.

If a positive integer n is so large that

\frac{4}{μ(E′)^{2}} ∑_{i≠j^{i,jϵℤ}} |χ̂_{E′} (m_{i} − m_{j})|^{2} ≤ n + 1

for each subset E′ ⊂ E of measure μ(E′) ≥ \frac{μ(E)}{2}, then there exists a set E_{1} ⊃ E such that

1) μ(E_{1}\E) ≥ \frac{μ(E)}{4n}

2) ∫_{E_{1}} |f|^{2}dμ ≤ \{\frac{An^{5}}{μ(E)^{2}} \}^{3n} ∫_{E} |f|^{2}dμ with some absolute constant A > 0.

This lemma was one of the most complicated results in [1] and to outline its full proof, one should rewrite about a half of 60 pages of [1]. So I have to refer the reader to [1] for details. Nonetheless, once having been proved, it is very easy to apply: we are merely to iterate its result until the measure of the next set E_{k}, say, will get greater than \frac{1}{2} and after that to use the statement of theorem 1. The main problem is to make the parameter n as small as possible at each step, i.e. to get a good estimate of the sum ∑_{i≠j} |χ̂_{E′} (m_{i} − m_{j})|^{2}.

In [1] the trivial estimate

\begin{matrix}∑_{i≠j} |χ̂_{E′} (m_{i} − m_{j})|^{2} = ∑_{r≠0} |χ̂_{E′} (r)|^{2}card \{(i, y) : m_{i} − m_{j} = r\} ≤ \\ R(Λ) ∑_{rϵℤ} |χ̂_{E′} (r)|^{2} = R(Λ)μ(E′) \end{matrix}

was used, which allowed to take n ∼ \frac{1}{μ(E)}. If we take into account that only rϵDΛ may occur in the last sum and that for any Λ(p)−system U ⊂ ℤ

\{ ∑_{rϵU} |χ̂_{E′} (r)|^{2}\}^{\frac{1}{2}} = sup \{| ∫_{E′} \overline{g}dμ| : ∥g∥_{L^{2}(𝕋)} ≤ 1, spec g ⊂ U\} ≤

≤ μ(E′)^{1− \frac{1}{p}} sup\{∥g∥_{L^{p}(𝕋)} : ∥g∥_{L^{2}(𝕋)} ≤ 1, spec g ⊂ U\} ≤ Cμ(E′)^{1− \frac{1}{p}} , we shall get much better inequality

∑_{i≠j} |χ̂_{E′} (m_{i} − m_{j})|^{2} ≤ R(Λ)C^{2}μ(E′)^{2− \frac{2}{p}} ,

which makes possible to put n = [\frac{B}{μ(E)^{2/p}} ] with some B > 0 depending on Λ only ([x] stands here for the entire part of a real number x, as usual).

Thus, if we denote by C(ν) the best possible constant, for which the inequality ∥f∥_{L^{2}(T)}^{2} ≤ C(ν) ∫_{E} |f|^{2}dμ holds for every function f ∈ L^{2}(𝕋) spectred at Λ and for every

set E ⊂ 𝕋 of measure μ(E) ≥ ν, and put ψ(ν) =^{def} log C(ν), Δ(ν) =^{def} \frac{1}{4B} ν^{1+ \frac{2}{p}}, we shall get

\frac{ψ(ν) − ψ(ν + Δ(ν))}{Δ(ν)} ≤ \frac{12B^{2}}{ν^{1+ \frac{4}{p}}} log \frac{AB^{5}}{ν^{2+ \frac{10}{p}}} ≤ \frac{B_{ε}}{ν^{1+ \frac{4}{p} +ε}}

for each 0 < ν ≤ \frac{1}{2}. Comparing this inequality with the corresponding differential that \frac{dψ}{dν} (ν) ≤ _{\frac{B_{ε}}{ν^{1+ \frac{4}{p} +ε}}}, we can easily conclude that ψ(ν) = O(ν^{− \frac{4}{p} −ε}) as it has been promised. It is well-known that a spectrum Λ with Hadamard gaps does actually satisfy the conditions of the theorem 2 with arbitrarily large p > 0 and, therefore, we get the estimate

∥f∥_{L^{2}(𝕋)}^{2} ≤ exp \{\frac{B_{ε}}{μ(E)^{ε}} \} ∫_{E} |f|^{2}dμ

with arbitrarily small ε > 0 and some B_{ε} > 0 depending on ε and Λ only (not on f or E).

Moreover, we can make p dependent on μ(E) when choosing n and obtain the better inequality.

∥f∥_{L^{2}(𝕋)}^{2} ≤ exp \{B log^{10} \frac{2}{μ(E)}\} ∫_{E} |f|^{2}dμ.

The power 10 isn’t, of course, the best possible one, but it cannot be replaced by any number less than 2 even for arcs E.

Another well-known exmaple, underlying the conditions of the theorem 2, is the following. Let p be an even integer great than 2, Λ = {m_{k}}_{k∈ℤ} be such that there is no nontrivial linear dependence between integers m_{k} with integer coefficients for which the sum of the absolute values of the coefficients is 2p or less (actually this is an example of a spectrum for which the corresponding constant, connecting L^{p} and L^{2}-norms of a function spectred at the difference set, is the least possible one among those for spectra with infinitely many elements). Such a spectrum can be rather dense, namely it can contain about N^{\frac{1}{2p} −ε} elements among first N successive integers (to construct the corresponding example, it is enough to consider a random spectrum Λ = {n ∈ ℕ : ξ_{n} = 1}, where ξ_{n}(n = 1, 2, ⋯) are independent random variables defined by P \{ξ_{n} = 1\} = δn^{\frac{1}{2p} −ε−1}, P \{ξ_{n} = 0\} = 1 − δn^{\frac{1}{2p} −ε−1} with δ > 0 sufficiently small). According to the Mandelbrojt-Belov-Miheev theorem, this means that the power −\frac{4}{p} − ε in the conclusion of the theorem 2 cannot be replaced by −\frac{1}{2p−1} + ε even for arcs E (at least for p = 4, 6, ⋯). The result of theorem 2 may be used in different ways. We shall mention below only a couple of the most immediate consequences; in both of them we suppose Λ to satisfy the conditions of the theorem 2 with some p > 4.

Corollary 3.

Let ā = {a_{k}}_{k=−∞}^{∞} be a sequence of complex numbers for which ∑_{k=−∞}^{∞} |a_{k}|^{2} < +∞ and

a_{k} = 0 for every k ∉ Λ. Let S be the left shift operator in ℓ^{2}(ℤ) (i.e. (Sa)_{i} = a_{i+1}, i ∈ ℤ). Then a is not a cyclic vector for S.

\underline{Proof}:

This is merely a reformulation of the obvious statement that either a = 0 or

∫_{𝕋} | log | ∑_{k∈ℤ} a_{k}z^{k}||dμ(z) < +∞.

\underline{Corollary 4}.

Let Λ ⊂ ℤ_{+} and f(z) = ∑_{k∈Λ} a_{k}z^{k} be an analytic function in the unit disk 𝔻.

(a) if ∑_{k∈Λ} |a_{k}|^{2} < +∞ then f ∈ H^{2}(𝔻) and has no singular factor in its canonical factorization,

(b) If ∑_{k∈Λ} |a_{k}|^{2} = +∞ then f has infinitely many zeros in 𝔻, these zeros being everywhere dense near the boundary of 𝔻 and “equidistributed” there in the following sense: let

τ(r)^{2} =^{def} ∑_{k∈Λ} |a_{k}|^{2} r^{2k} (0 < r < 1), ν_{r} =^{def} ∑_{0<|ξ|<r^{ξ:f(ξ)=0}} log \frac{r}{|ξ|}δ_{ξ}

(as usual δ_{ξ} denotes the unit mass concentrated at ξ and zeros are counted with their multiplicity); then \frac{1}{log τ(r)}ν_{r} weakly converges to the Lebesgue measure μ on the unit circumference when r → 1−.

\underline{Remark.}

The statement (b) is worth comparing with the well-known result due to Murai claiming that for spectra Λ ⊂ ℤ_{+} with Hadamard gaps the density of zeros near the boundary may be derived from much weaker assumption sup_{z∈𝔻} |f(z)| = +∞ (or, what is the same for series with Hadamard gaps, ∑_{k∈Λ} |a_{k}| = +∞). But Murai’s proof uses the Hadamard lacunarity condition in its full strength and I don’t know whether there is any essentially denser spectrum Λ for which his result remains valid.

\underline{Proof}:

The part (a) follows immediately from the simple observation that either f ≡ 0 or the integrals ∫_{𝕋} | log |f(rz)∥^{1+δ} dμ(z) are uniformly bounded for \frac{1}{2} < r < 1 if δ > 0 is sufficiently small.

To prove the statement (b) it is enough to check that the measures η_{r} =^{def} \frac{1}{log τ(r)}ν_{r} satisfy the following two conditions:

1) for every R < 1 η_{r}(𝔻_{R}) → 0 when r → 1− (here 𝔻_{R} =^{def}{z : |z| ≤ R}.

2) if |ξ| < 1 and f(ξ) ≠ 0, then ∫_{C} \frac{1−|ξ|^{2}}{|z−ξ|^{2}} dη_{r}(z) → 1 when r → 1−

(the set of linear combinations of Poisson kernels is dense in C(𝕋)).

(1) is obvious because there are only finitely many zeros of f within 𝔻_{R} and τ(r) → +∞ when r → 1−. When checking (2) we may assume without loss of generality that f(0) ≠ 0 (the division of f by z^{m} changes nothing). If r > |ξ| we can write the Jensen formula

∑_{{ζ:|ζ|<r \atop f(ζ)=0}} log \frac{|r^{2} − \overline{ξ}ζ|}{r|ξ − η|} = ∫_{𝕋} log |f(rz)|\frac{r^{2} − |ξ|^{2}}{|ξ − rz|^{2}} dμ(z) − log |f(ξ)|.

The result of the theorem 2 implies immediately that the right hand part is log τ(r)+O(1) when ξ is fixed and r → 1−.

Therefore

∫_{D} log \frac{|r^{2} − \overline{ξ}z|}{r|z − ξ|} dη_{r}(z) → 1 when r → 1 − .

Note now that

log \frac{|r^{2} − \overline{ξ}z|}{r|z − ξ|} = \frac{1}{2} log (1 + \frac{r^{2} − |ξ|^{2}}{|ξ − z|^{2}} (1 − |\frac{z}{r} |^{2})) .

The ratio \frac{r^{2}−|ξ|^{2}}{|ξ−z|^{2}} is uniformly bounded for all z ∈ supp η_{r} (i.e. for z satisfying f(z) = 0); and the ratio |\frac{z}{r} | is very close to 1 when r is close to 1 and |z| is close to 1. Thus we can replace log \frac{|r^{2}−\overline{ξ}z|}{r|z−ξ|} by the equivalent expression \frac{1−|ξ|^{2}}{|z−ξ|^{2}} log|\frac{r}{z} |, getting the result desired. Very similar considerations allow also to prove the following Corollary 5.

Fix any (small) δ > 0 denote T_{δ} =^{def}{x + iy : 1 − δ < x < 1, y < δ(1 − x)}. If f satisfies the conditions of the corollary 4 then f(ζT_{δ}) is dense in ℂ for almost all ζ ∈ 𝕋. Literature cited: [1] [2] [3]