Zero-free neighborhood around the unit circle for Kac polynomials

In this paper, we study how the roots of the so-called Kac polynomial $W_n(z) = \sum_{k=0}^{n-1} \xi_k z^k$ are concentrating to the unit circle when its coefficients of $W_n$ are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial are concentrating around the unit circle as $n\to\infty$ if and only if $\mathbb{E}[\log( 1+ |\xi_0|)]<\infty$. Under the condition of $\mathbb{E}[\xi^2_0]<\infty$, we show that there exists an annulus of width $\mathrm{O}(n^{-2}(\log n)^{-3})$ around the unit circle which is free of roots with probability $1-\mathrm{O}({(\log n)^{-{1}/{2}}})$. The proof relies on the so-called small ball probability inequalities and the least common denominator.


Introduction
In the signal analysis and the speech recognition, an important tool is the socalled z-transform (a particular case is the discrete Fourier transform). In this context, the study of zeros of the z-transform provides useful information of a signal. The existence of a region free of zeros around the unit circle is an important aspect for a proper work of the z-transform. For further details, see Chapter 3 in [4]. Roughly speaking, in this paper we find a region free of zeros around the unit circle for the so-called Kac polynomials which are closely related to the discrete Fourier transform.
Let n ∈ N and let ξ 0 , . . . , ξ n−1 be independent and identically distributed (iid for shorthand) non-degenerate real random variables (rvs for shorthand) defined in the probability space (Ω, F , P). Denote by E the expectation respect to the measure P. The Kac polynomial W n is defined as the random polynomial of degree n − 1 given by W n (z) = n−1 j=0 ξ j z j , z ∈ C.
In the sequel, we introduce the basic notation and terminology which will be used throughout this manuscript. For any z ∈ C, denote by |z| and arg(z) the modulus of z and the argument of z, respectively. Let a, b ∈ R such that a ≤ b. Denote by R n (a, b) the number of roots of W n in the annulus {z ∈ C : a ≤ |z| ≤ b} and for any α, β ∈ [−π, π] such that α ≤ β denote by S n (α, β) the number of roots in {z ∈ C : α ≤ arg(z) ≤ β}.
Shparo and Shur [19] proved that under general conditions on the random coefficients (rcs for shorthand), the roots of W n concentrate around the unit circle with asymptotically uniform distribution in the argument as n goes by. Moreover, Ibragimov and Zaporozhets [7] showed that if the rcs of W n are non-degenerate and satisfy E[log(1 + |ξ 0 |)] < ∞ if and only if its roots are asymptotically concentrated near the unit circle. Later, Kabluchko and Zaporozhets [9] provided a wide description of the localization of the roots for different conditions on the rcs. We point out that the localization of the roots of Kac polynomials determine the poor efficiency of some algorithms from the speech recognition and signal processing applications, see [5] for further details.
In the case that W n has iid rcs which common distribution belongs to the domain of attraction of an α-stable law, limit (2) yields that for δ > 0, W n has at least one root in the annulus R δ,n := {z ∈ C : e −δ/n ≤ |z| ≤ e δ/n } with positive probability for all large n and Therefore, a remarkable question is to determine if there exists an annulus inside of R δ,n such that W n has at least one root or not on it. The existence of roots pretty close to the unit circle is an important aspect in the analysis of signals. This helps to understand the contribution of the phase information of a signal. We refer to [5] for further details. Shepp and Vanderbei [18] conjectured that with high probability the nearest root of W n to the unit circle is at a distance of order O(n −2 ). Later, Konyagin and Schlag [12] showed that the last conjecture holds true when the rcs have standard Gaussian or Rademacher (uniform distribution on {−1, 1}) distribution. To be more precise, Konyagin and Schlag proved that there exists a positive constant C such that for any t > 0 They also showed (4) P min for γ > 1/2 and iid Gaussian rcs. Karapetyan [11] mentioned that it is possible to extend the above result under the assumption of non-degenerate real sub-Gaussian rcs, but only a sketch of the proof was given. Moreover, he claimed that the previous result can be extended under the finiteness of the third moment on the rcs. However, Karapetyan [10] showed that for iid rcs with zero mean and finite third moment, it follows for any ǫ ∈ (0, 1) and n > 16C P min where the constant C depends only on the moments of ξ 0 . The proof of (5) is long, technical and complicated. Later, Barrera and Manrique [2] proved that if the moment generating function of iid coefficients exists in an open neighborhood around 0, then for any t ≥ 1 where γ > 1/2. The proof of (6) recovers the essential ideas of Konyagin and Schlag [12] who only considered the problem when the rcs have Rademacher or standard Gaussian distribution. Their proof is quite technical and involved. It is based on the so-called Salem-Zygmund inequality for sub-Gaussian rvs.
To extend for more distributions, Barrera and Manrique [2] took advantage of the concept of least common denominator (lcd for shorthand), which was developed in the study of the singularity of random matrices [16]. Roughly speaking, the lcd is a combinatorial measurement to understand the concentration of a sum of independent rvs in a small ball. Furthermore, under the assumptions of the finiteness of the second moment, using similar ideas from Barrera and Manrique [2], it is possible to find an annulus in which W n does not have roots with high probability. We remark that the lcd has been converted into a useful tool that allows analyzing different interesting problems. For instance, it is used in the study of isomorphism between graphs [14] and in the analysis of the condition number for random matrices [17]. In this manuscript, the lcd is used to understand how small can be the modulus of a random polynomial near the unit circle.
In this work, the lcd allows us to develop clear arguments to estimate how close are the roots of a Kac polynomial to the unit circle. To be more precise, when the rcs of a Kac polynomial are iid rvs with zero mean and finite second moment, the majority of the roots are a distance of order O(n −2 (log n) −3 ) with probability 1 − O((log n) −1/2 ). The main obstacle to extend this result comes from the Salem-Zygmund inequality as we will see in Section 2.
The main result of this paper is the following. Theorem 1.1. Let {ξ j : j ≥ 0} be a sequence of real iid non-degenerate real rvs satisfying for some M > 0, γ > 0 and q ∈ (0, 1). Assume that E[ξ 0 ] = 0 and E[ξ 2 0 ] < ∞. Then for all t ≥ 1 fixed we have where the implicit constant depends on t and the distribution of ξ 0 . Remark 1.2.
(1) In Theorem 1.1 only is assumed the finiteness of the second moment, zero mean and condition (H) which include Rademacher and standard Gaussian rvs. As a directly consequence of Theorem 1.1, we have (2) We point out that in (7) we consider the minimum of the modulus of the Kac polynomial over the set {||z| − 1| ≤ tn −2 (log n) −3 } which is properly contained into the region considered in expression (3), but it contains the region considered in (4). Nevertheless, we obtain the upper bound O((log n) −1/2 ) which improves the bound given in (3).
This manuscript is organized as follows. In Section 2 we give an outline of the proof. In Section 3 we provide the proof of Theorem 1.1. Finally, in Appendix A we prove auxiliary results that we used throughout the manuscript.

Outline of the proof
In this section, we present the strategy used to prove Theorem 1.1. Our goal is to estimate P(A n ), where and t ≥ 1 is a fix constant. First, motivated by the estimates given in [12], Section 2, p. 4964, we analyze the probability of the events where N n and g n are appropriate functions of n (later on, we provide precise description of them). We anticipate that N n ≈ n 2 (log(n)) 3 , which is similar to the number of balls using in [12]. We point out that N n needs to trade off g n in order to the probability of A n,α tends to zero, as n → ∞. Second, for each α = 0, . . . , N n − 1 we analyze the arithmetic structure of the sequence {exp(i2πjx α ) : j = 0, . . . , n − 1} and using the so-called small ball inequalities we prove that P(A n,α ) → 0, as n → ∞. The idea is to apply the Taylor Theorem to approximate W n in small balls with centers at exp(i2πx α ). It allows us to write the event A n as the union of events of the form A n,α . However, we need to handle the maximum value for the derivative of W n on the unit circle. The latter can be done by a Salem-Zygmund type inequality, which consists in estimate the maximum possible value of a Kac polynomial on the unit circle.
Denote by W ′ n ∞ the supremum norm of W ′ n over the unit circle. In the case of ξ 0 , . . . , ξ n−1 being iid sub-Gaussian rvs, a Salem-Zygmund type inequality (in probability) gives for some suitable positive constant C p . In [2], the authors showed that (8) holds for iid zero mean rvs with finite moment generating function. In this paper, we are not assuming the existence of the moment generating function. Instead of, we assume finiteness of the second moment. By applying the majorizing measure method, Weber [20] showed (8) in expectation.
To be more precise, let ξ 0 , ξ 1 , . . . , ξ n−1 be iid zero mean rvs with finite second moment, Corollary 2 in [20] implies that there exists a positive constantC (only depends on To improve Theorem 1.1 (using lcd technique) to more general rcs we require a refined version of the Salem-Zygmund inequality for rvs without finite second moment. At the moment, the authors are not able to obtain a Salem-Zygmund type inequality for rvs without the finiteness of the second moment. Later, we apply small ball inequalities to show that This kind of inequalities allow us to consider more general rcs and provide a new proof of the main theorem in [12]. To apply small ball inequalities we analyze the lcd for some specific matrix. In the sequel, we give the definition of the lcd for a matrix. Denote log + x := max {log x, 0} for any x > 0.
Definition 2.1 (Least common denominator (lcd)). Let L > 0 be a positive number. Let · 2 be the standard Euclidean norm and let dist v, Z M denote the distance between the vector v ∈ R M and the set Z M . For a given matrix V ∈ R m×M the lcd is defined as For a review of the concept lcd, we recommend Section 7 in [16]. For our purposes, in Definition 2.1 we take m = 2, M = n and the matrix V is given by Note that if det V V T > 0, Theorem 7.5 in [16] implies that for a > 0 and t ≥ 0 where L ≥ 2/q and the constant C only depends on constants M , γ, q specified in Theorem 1.1. By Definition 2.1 it is not hard to deduce that for any a > 0, D(aV ) ≥ ( 1 /a)D(V ). Recall the inequality (x + y) 2 ≤ 2x 2 + 2y 2 for any x, y ∈ R. By (9) we deduce that Nn , the arithmetic properties of x α given by α and N n should play an important role in the estimates. Depending on the greatest common divisor between α and N n , gcd (α, N n ), we deduce suitable positive lower bounds for det V T V and dist V T Θ, Z n which together with (K) allow us to show that P ( V X 2 ≤ g n ) is sufficiently small.
Taylor's approximation. In the sequel, define the trigonometric random polynomial T n (x) := n−1 j=0 ξ j e ijx , x ∈ R and denote by T ′ n its derivative with respect to x. To do the notation shorter, we denote by ∆ n the following event: where C 0 is a positive constant that we will precise later. We also denote by P (A, B) the probability P (A ∩ B) for any two events A and B. By the total probability law, we deduce  [15]) allows us to deduce for the second term in the right-side hand of (10) that P T ′ n ∞ > C 0 n 3/2 log n ≤ P T n ∞ > C 0 n 1/2 log n .

Remark 2.2.
In the preceding reasoning we only use zero mean and finiteness of the second moment of ξ 0 . In particular, it holds for sub-Gaussian rvs which includes Rademacher, standard Gaussian, and bounded rvs.
Arithmetic properties of x α . In the sequel, we decompose the event A n ∩ ∆ n into regions for which the arithmetic properties of x α are useful in order to use the anti-concentration assumption (H) and allows us to show P(A n , ∆ n ) tends to zero, as n → ∞. We point out that in the following reasoning we only use assumption (H). To achieve our goal, we consider a set of balls with center at points on the unit circle with an adequate radius. We distinguish two kind of balls. The special balls with center at 1 + 0i and −1 + 0i, where the radius r is large (r = 2tn −11/10 ), and the balls with center at points z with argument satisfying n −11/10 < |arg(z)| < π − n −11/10 and small radius (r = 2tn −2 (log n) −3 ). Recall that for any x ∈ R, ⌊x⌋ denotes the greatest integer less than or equal to x. Let N = ⌊n 2 (log n) 3 ⌋ and x α = α N for α = 0, 1, . . . , N − 1. For a ∈ C and s > 0, denote by B (a, s) the closed ball with center a and radius s, i.e., B (a, s) = {z ∈ C : |z − a| ≤ s}. Denote by S 1 the unit circle. Let A S 1 , tn −2 (log n) −3 := z ∈ C : ||z| − 1| ≤ tn −2 (log n) −3 .
The preceding inclusion yields that any z ∈ A with small argument belongs in the union of the balls with center at 1 + 0i and −1 + 0i with radius 2tn −11/10 . On the other hand, for z ∈ A with large argument we have where gcd(α, N ) denotes the greatest common divisor between α and N . For any α ∈ J 3 (n, N ) we have The preceding inequalities mean that the irreducible fraction of x α is as small as a multiple of n −11/10 . Therefore, We emphasize that if α ∈ J 1 (n, N ) ∪ J 2 (n, N ) ∪ J 3 (n, N ). Then we have n −11/10 < |2πx α | < π − n −11/10 .
Lemma 2.5. Assume that n 11/10 Then for a suitable constantC it follows α∈J2(n,N ) where the implicit constant in the big O-notation depends on L and t.
In the sequel, we stress the fact that Theorem 1.1 is just a consequence of what we have already stated up to here. Indeed, combining Lemma 2.3, Lemma 2.4, Lemma 2.5, Lemma 2.6, estimate (11) and estimate (12) in inequality (10) yield Theorem 1.1.

Proof of Theorem 1.1
In this section, we show that the left-hand side of inequality (13) is of order O((log(n)) −1/2 ).

3.1.
Estimates on the balls centered at −1 and 1.

3.2.
Estimates of P (∆ n , B α ). In the sequel, we apply the Taylor Theorem repeatedly in order to reduce P (∆ n , B α ) to an estimate of the probability of how small a sum of iid rvs can be. The latter can be computed (estimated) using small ball inequalities. Let z ∈ B e i2πxα , 2tn −2 (log n) −3 and assume that ∆ n holds. The Taylor Theorem yields where o(1) = 2tn −2 (log n) −3 . Then To show that P (∆ n , B α ) tends to zero as n increases, we rewrite the sum W n e i2πxα as the product of a matrix by a vector, and then we analyze the lcd of the corresponding matrix. Define the 2 × n matrix V α as follows and take X = [ξ 0 , . . . , ξ n−1 ] T ∈ R n . Notice that Let Θ = r [cos (θ) , sin (θ)] T ∈ R 2 , where r > 0 and θ ∈ [0, 2π). For fix r and θ we have We also point out that V T α Θ 2 ≤ r √ n. On the other hand, we observe that Bearing all this in mind, we can use the notion of lcd for high dimensions to obtain an accurate upper bound of the left-hand side of (14).
Therefore, the preceding inequality together with inequality (K), inequality (18) and the fact the cardinality of J 2 (n, N ) is at most N allow us to deduce α∈J2(n,N ) for all large n, where c 4 is a positive constant. As a consequence we obtain α∈J2(n,N ) where the implicit constant depends on L and t.

Estimates on J 3 (n, N ).
Proof of Lemma 2.6. This case requires a more refined analysis. Observe that where o (1) = 1 n 2 (log n) 3 . Then 2πx α satisfies By Lemma 3.2 Part 2 in [12], there exist positive constants c 4 , C 4 such that (19) c 4 n 2−1/5 ≤ det V α V T and for each j ∈ I r M , we define Note that Let L = min n 8r − 3 2 , N ′ 8r − 1 2 and observe that for each 0 ≤ λ ≤ L there exists at least j ∈ I r M such that d j ≥ (2λ + 1) 2π N ′ . Then By the choosing of L, if r ≤ n 1/4 , we have 2πL 2 N ′ ≥ 2π n 11/10 n 3/2 for all large n. Here, let v be a vector in R n with entries v j = cos (j2πx α − θ) for each j = 0, . . . , n − 1. If a positive integer r ≤ n 1/4 , then by the previous discussion we deduce dist(rv, Z n ) ≥ 2πn 2/5 for all n large. If r is any positive real number, observe that s r , s+1 r ⊂ s ⌈r⌉ , s+1 ⌈r⌉ , where s ∈ N, and therefore our previous analysis holds true for any r > 0.
Assume that r ≤ n 1/4 and recall that V T α Θ 2 ≤ r √ n and that L ≥ 2/q is fixed. By the definition of lcd, for all n large we obtain which yields a contradiction for n large. Thus, D (V α ) ≥ n 1/4 . Therefore, the preceding inequality together with inequality (K), inequality (19) and the fact the cardinality of J 3 (n, N ) is at most n 11/10+o(1) (log(n)) o(1) allow us to deduce α∈J3(n,N ) P |W n e i2πxα | ≤ 2tC 2 n −1/2 (log n) Let L be the biggest integer such that 8πLx ≤ 1 2r , or equivalently, L = where the last inequality follows since 2r−1 r ≥ 1 for r ∈ N. Consequently, the distance between the vector V ∈ R n with entries V j = r cos (j2πx − θ) for j = 0, . . . , n − 1 with x = 1/n and the set Z n is at least verifying that 1 4πrx ≥ 8 is fulfilled.