CC BY
33
УДК 517.55
Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum
Received 10.05.2019, received in revised form 10.06.2019, accepted 20.09.2019 For f a nonzero Bohr almost periodic function on R with a bounded spectrum we proved there exist Cf > 0 and integer n > 0 such that for every u > 0 the mean measure of the set { x : \f (x)\ < u } is less than Cf u1/n. For trigonometric polynomials with < n + 1 frequencies we showed that Cf can be chosen to depend only on n and the modulus of the largest coefficient of f. We showed this bound implies that the Mahler measure M(h), of the lift h of f to a compactification G of R, is positive and discussed the relationship of Mahler measure to the Riemann Hypothesis.
Keywords: almost periodic function, entire function, Beurling factorization, Mahler measure, Riemann hypothesis.
DOI: 10.17516/1997-1397-2019-12-5-571-578.
1. Distribution of small values
Wayne M. Lawton*
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
N := {1,2,...}, Z, R, C, T := {z G C : \z\ = 1} are the natural, integer, real, complex and circle group numbers, Cb(R) is the C*-algebra of bounded continuous functions and xw : R ^ T, w G R the homomorphisms x u> (x) := eiwx, w G R. A finite sum f = ^ aw xw
with distinct w is called a trigonometric polynomial with height Hf := maxw \aw \ and they comprise the algebra T(R) of trigonometric polynomials. Bohr [9] defined the C*-algebra U(R) of uniformly almost periodic functions to be the closure of T(R) in Cb(R) and proved that their means m(f) := limL^TO(2L)-1 fLL f (t)dt exist. The Fourier transform f : R ^ C of f G U(R) is f(w) := m(fxw ) and its spectrum Q(f) := support f. If f is nonzero then Q(f) is nonempty and countable and we say f has bounded spectrum if its bandwidth b(f) G [0, to], defined by b(f) := sup^(f) — inf Q(f), is finite. We observe that if S C R is defined by a finite number of inequalities involving functions in U(R) then m(S) := limL^TO(2L)-1measure —L, L] n S exists and define Jf : (0, to) ^ [0,1] by
Jf (u) := m ( { x G R : \f (x)\ <u } ) (1)
Theorem 1.1. If f G U(R) is nonzero and has a bounded spectrum then there exist Cf > 0 and n G N such that:
Jf (u) < Cf un, u > 0. (2)
There exists a sequence Cn such that if f G T(R) has n +1 frequencies then
i
Jf (u) < Cn Hfn un, u> 0. (3)
* [email protected] © Siberian Federal University. All rights reserved
Proof. For f G U(R), w G R,k G N, u > 0 define , Kf : (0, to) ^ [0,1] by
S^mM- x e R : f (x)| < u, |( Xw f )(j)(x) | < vj ,j = 1,...,k }, (4)
Kf (u) := inf inf inf [3 V2n-1 b(f) kv-1 uk + Sf w k u(v)l . (5)
' weR keN v>0 L J
We first prove Theorem 1 assuming the following result which we prove latter. Lemma 1.1. Every nonzero f G U(R) with bounded spectrum satisfies Jf ^ Kf.
We observe that for every w G R and every a G R\{0}, if h(x) = xw(x) f (ax) then Jh = Jf
r hi hi ]
and Kh = Kf. Without loss of generality we can assume that tt(f) C — , . If b(f) = 0 then f = c and Jf (u) < lcl-1u. If b(f) > 0 then Bohr [10] proved that f extends to an entire function F of exponential type ^^, and Boas [6], ([7], p. 11, Equation 2.2.12) proved that
limsup \f (k)(x)\k = f
2 (6) hi
uniformly in x. Therefore for any v0 > —— there exists k G N such that Sf,0 ,k u(v0) = 0 so
Lemma 1.1 implies Jf satisfies (2) with Cf = 3 %/2n-1 b(f) kv-1 and n = k. This proves the first assertion. To prove the second we assume, without loss of generality, that b(f) = 1, Q(f) C [0,1] and
n+1
f (x) = X/ aj e%Uj x, 0 = w1 < ■■■ <wn+1 = 1, Hf =max{|aj | : j = 2,...,n + 1}.
j=1
Define C1 := 2. If n =1 and f has n +1 = 2 terms and f = a0 + a1 x1 with |ai | = Hf and h = Hf (1 - x1), then Jf (u) < Jh(u) = (2/n)sin-1(u/(2Hf)) < C1H-1 u therefore (3) holds for n =1. For n > 2 we assume by induction that (3) holds for n — 1 and therefore, since f(1) has n terms and Hf (1) = Hf, it follows that for all v > 0,
1
Jf (1) (v) < Cn-1 HJ-1 v, (7)
1
Sf,0,1,u(v) < Cn-1 HJ-1 v^. (8)
Therefore Lemma 1 with w = 0, b(f) = k =1 gives
Jf (u) < inf
' K ' v>0
3 V2 n-1 v-1 u + Cn-lnf-1
n-lHf ' v
Cn Hf un (9)
where Cn := C—l [3V2 n-1 (n - 1)]n n(n - 1)-1. (10)
Remark 1.1. Computation of 200 million terms shows that n-1Cn ^ 0.900316322 Conjecture 1.1. In (3) Cn can be replaced by a bounded sequence.
Lemma 1.2. If $ : [a,b] ^ C is differentiable and $'([a,b]) is contained in a quadrant then
^ max \$\([a, b])
b - a < W2 . (11)
min \([a, b])
Proof of Lemma 1.2. We first proved this result in ( [18], Lemma 1) where we used it to give a proof, of a conjecture of Boyd [11] about monic polynomials related to Lehmer's conjecture [20], which was reviewed in ([13], Section 3.5) and extended to monic trigonometric polynomials in ([19], Lemma 2). The triangle inequality < + ^gives
/• b fb
(b — a) min W\([a,b]) ^ \J>'(y)\dy ^ (J>'(y)\ + \S J>'(y)\) dy.
J a J a
Since ^>'([a, b]) is contained in a quadrant of C there exist c,d G {1, —1} such that ^(p'(y)\ = cK4>'(y) and \S^'(y)\ = dS$'(y) for all y G [a,b]. Therefore
I ( 4>'(y)\ + \S 4>'(y)\) dy = (cK^(b)+ d$4>(b)) — (c^(a)+ d$4>(a)).
a
The result follows since the right side is bounded above by 2%/2 max \^\([a, b]). □
Proof of Lemma 1.1, Assume that f G U(R) is nonzero. We may assume without loss of generality that Q,(f) C [— f, f ]. For k G N,u > 0, v > 0 we define the set
Sf,k,u,v := {x G R : \f(u)\ < u, max \f(j)(x)\1 > v }. (12)
je{i,...,h}
We observe that the set of functions in U(R) whose spectrums are in [— , ] is closed under differentiation, and define s(f,k,u,v) := m(Sf,k,u,v).
It suffices to prove that s(f,k,u,v) < 3%/2n-1 b(f) kv-1 u1. (13)
Define Yj := ukr vj,j G {0, ...,k}, and I := set of closed intervals I satisfying, for some j G {0,1, ...,k — 1}, the following three properties:
1. f (j+1)(i) is a subset of a closed quadrant,
2. max \f(j)\(I) < Yj and min \f(j+1)\(I) > Yj+1,
3. I is maximum with respect to properties 1 and 2. Define E := set of endpoints of intervals in I, and
k-1
* :=H(Mf(j+1))(Sf(j+1))(\f(j)(x)\2 — y])(\f(j+1)(x)\2 — Yj2+1). (14)
j=0
Lemma 1.2 implies that length (I) < = 2V2v-1 uk, I GI, (15)
-j a-J aij )(x)\2 — Y2 )
j=0
Yk = 2./2 v-1, Yk+1
and (12) and Property 3 implies that Sf,k,u,v C I. (16)
I€I
Clearly * = where ^ is the product of 6k entire functions each having bandwidth b(f) so a theorem of Titchmarsh [25] implies that the density of real zeros of ^ is bounded above by 3n-1 b(f) k. Property 3 implies that all points in E are zeros of ^ so the upper density of intervals in I is bounded by ] n-1 b(f) k. Combining these facts gives s(f,k,u,v) < (3n-1 b(f) k) (2v/2v-1 uk) = 3v/2n-1 b(k) kv-1 u* which proves (13) and concludes the proof of Lemma 1. □
Forp G [1, to) Besicovitch [4] proved that the completion Bp(R) of U(R) with norm (m(\f \p)) p
loc (
to, 0], and \x\^ := max{\x\, j
is a subset of Lfoc(R). For x > 0 we define log+(x) := log(max{1, x}) G [0, to), log (x) := log(min{1, x}) G [—to, 0], and \x\j :=max{\x\,1} for j G N.
Corollary 1.1. If f e U(R) satisfies (2), then log" o \f \ e Bp(R),
10
and log o\f \ e Bp(R).
m( \ log" o\f \\p ) < f1 \ log(u)\p dCf un = Cf np r(p), (17)
0
Proof of Corollary 1.1. Since the means of the functions log" o\fj \p are nondecreasing and bounded by the right side of (17), the sequence log" o\fj is a Cauchy sequence in Bp(R) so it converges to a function n € Bp(R). Therefore log" o\f \ = n since it is the pointwise limit of log" o\f\j and n € Lpoc(R). The last fact follows since log = log+ +log" . □
2. Compactifications and Hardy Spaces
Definition 2.1. A compactification of R is a pair (G,6) where G is a compact abelian group and 6 : R ^ G is a continuous homomorphism with a dense image.
C(G) is the set of continuous functions on G and Lp(G),p e [1, to) are Banach spaces. If h e C(G) then f := h o 6 e U(R) since by a theorem of Bochner [8] every sequence of translates of f has a subsequence that converges uniformly. We call h the lift of f to G. The Pontryagin dual [24] G of a compact abelian group G is the discrete group of continuous homomorphisms X : G ^ T under pointwise multiplication. Bohr proved the existence of a compactification (B, 6) such that U(R) = { h o 6 : h e C(B) }. The group B is nonseparable and B is isomorphic to R d := real numbers with the discrete topology.
Lemma 2.1. For every f e U(R) there exists a compactification (G(f), 6), with G(f) separable, and h e C(R) such that f = h o 6.
Proof of Lemma 2.1. If f e U(R) is nonzero its spectrum Q(f) is nonempty and countable so the product group TQ(f) is compact and separable. The function 6 : R ^ TQ(f) defined by 6(x)(w) := Xw(x) is a continuous homomorphism. Define G(f) := 6(R). Then (G(f),6) is a compactification. The function h : 6(R) ^ C defined by h(6(x) := f (x) is uniformly continuous so extends to a unique function h : G ^ C and f = h o 6. □
Lemma 2.2. If (G,6) is a compactification, h e C(G), f = h o 6, and log o\f \ e Bp(R), then log o\h\ e Lp(G) and fG \ log o\h\ \p = m(\log o\f \ \p) for all p e [1, to).
Proof of Lemma 2.2. The theorem of averages ( [3], p. 286) implies that
f \ log" o\h\j \p = m(\log" o\f\j \p) < m(\log" o\f\\p). (18)
JG
The result follows from Lebesgue's monotone convergence theorem since the sequence \ log o\h\j \p is nondecreasing, converges pointwise to \ logo\h\ \p pointwise and by (18) their integrals are uniformly bounded. □
Definition 2.2. The Fourier transform F : L1(G) ^ (^(G) is defined by F(h)(x) := / fx.
G
We define the spectrum Q(h) := support F(h). The Hausdorff-Young theorem [15,26] implies that the restrictions give bounded operators F : Lp(G) ^ lq(G) for p e [1, to) and p"1 +q"1 = 1.
Definition 2.3. A compactification (G, 9) induces an injective homomorphism £ : G ^ R, £(x) := w where x ◦ 9 = xu, by which we will identity G as a subset of R with the same archimedian order. Therefore if h G C(G) is the lift of f G U(R), then Q,(h) = Q,(f). The compactification gives Hardy spaces Hp(G, 9) := {h G Lp(G) : Q,(h) C [0, to)}, p G [1, to].
Definition 2.4. A function h G Hp(G, 9) is outer if J h = 0, log o\h\ G L1(G), and
a
log o\h\ = log
Gh <G
(19)
A function h G Hp(G, 9) is inner if \h\ = 1.
A polynomial h is outer iff it has no zeros in the open unit disk since a formula of Jensen [16] gives /logo\h\ = log\h(0)\ — log-(\A\). Beurling [5] proved that a function h G H2(T)
G h(\)=0
admits a factorization h = ho hi, with ho outer and hi inner, iff log o \ h\ G L1(T).
Let (G, 9) be a compactification. If h G C(G) has a bounded spectrum Q(h) C [0, to) and f h da > 0 then f = h o 9 extends to an entire function F bounded in the upper half plane. We
G
observe that if F has no zeros in the upper half plane, then x-b(f)/2F is the Ahiezer spectral factor [1] of the entire function F(z)F(z).
Conjecture 2.1. h above is outer iff F has no zeros in the open upper half plane.
3. Mahler Measure and the Riemann Hypothesis
Definition 3.1. For G a compact abelian group the Mahler measure [22, 23] of h G L1(G) is M (h) := exp ^ / log o\h\^J G [0, to). We also define M±(h) := exp ^ / log± o\h\^J .
Since M(h) = M+(h)M-(h) and M +(h) G [1, max{1,\\h\U], it follows that M(h) > 0 iff log- o\h\ G L1(G) and then M-(h) = exp (—\\ log- o\h\ ^ ) . Lemma 2.2 implies that this condition holds whenever h G C(G) is nonzero and Q(h) is bounded.
Definition 3.2. For N G N, := product of the first N cyclotomic polynomials.
Amoroso ([2], Theorem 1.3) proved that the Riemann Hypothesis is equivalent to
logM ) <e N2+e, e > 0. (20)
Define fN := o x1 G U(R) and define JfN : (0, to) ^ [0,1] by (1). Jensen's formula implies that M($N) = 1 therefore
log M + ($N) = —I log(u) dJfN (u). (21)
0
The bounds that we obtained for Jf in (2) and (3) were exceptionally crude and totally inadequate to obtain (20). When deriving (3) for general polynomials we used the bound (8) Sfi0j1jU(v) = = m({ x : \f (x)\ < u, \f (1)(x)\ <v } < m({ x : \f (1)(x)\ <v }. Conjecture (1.1) was based on our intuition that a smaller upper bound holds. We suspect that much smaller upper bounds
G
hold for specific sequences of polynomials as illustrated by the following examples. Construct sequences of height 1 polynomials
Pn(z):=1 + z + ... + zn ; Qn (*):=( [^ ) (1 + ZT (22)
and pn := Pn o x1, qn := Qn o x1. Both polynomials have maxima at z = 1, \\Pn\\TO = n +1, Stirling's approximation gives \\Qn\\TO ~ \Jnn/2 for large n, and for u e (0,1]
2
Jpn (u) < - sin"1 (min{ 1, u } ) < u ^ log(M"(Pn)) > -1, (23)
2 ( i 1 / n \ " 1 1 \ 2 i 2n
Jqn (u) = - sin-M min<h, ^[n/2J u^j > nun ^ log(M "(Qn)) <-V. (24)
Differences between these polynomials arise from their root discrepancy. Those of Pn are nearly evenly spaced. Those of Qn, all at z = -1, have maximally discrepancy.
Conjecture 3.1. If Rn is a polynomial with n +1 terms and height H(Rn) = 1 then M"(Qn) ^ M"(Rn) < M"(Pn).
The roots of have the form exp(2niak), k = 1,..., deg where ak are the Farey series consisting of rational numbers in [0,1) whose denominators are < N. Bounds on the discrepancy of the Farey series were shown by Franel [14] and by Landau [17] to imply the Riemann Hypothesis. The relationship between the discrepancy of roots of a polynomial and its coefficients, and the distributions of roots of entire functions have been extensively studied since the seminal paper by Erdos and Turan [12] and the extensive work by Levin and his school [21]. We suggest that investigation of the functions Ef , w,k, v, u in (4) and derived functions Kf in (5) may further elucidate how the distribution of small values of polynomials and entire functions depend on their coefficients and roots.
The author thanks Professor August Tsikh for insightful discussions.
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Распределение малых значений почти периодических функций Бора с ограниченным спектром
Уэйн М. Лоутон
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Для f ненулевой почти периодической функции Бора на К с ограниченным спектром мы доказали, что существуют С/ > 0 и целое число п > 0 такие что для каждого и > 0 средняя мера установить { х : \f (х)\ < и } меньше С/ и1/п. Для тригонометрических полиномов с частотами < п + 1 мы показали, что С/ можно выбрать так, чтобы он зависел только от п и модуль наибольшего коэффициента Из этой оценки следует, что мера Малера М(Н), подъема Н из f к компактификации О из К положительна и обсуждена связь меры Малера с гипотезой Римана.
Ключевые слова: почти периодическая функция, целая функция, факторизация Берлинга, мера Малера, гипотеза Римана.
