Tamas Erdelyi - Profile on Academia.edu (original) (raw)

Papers by Tamas Erdelyi

arXiv (Cornell University), Feb 7, 2018

, where P k and Q k are the usual Rudin-Shapiro polynomials of degree n -1 with n = 2 k . In a re... more , where P k and Q k are the usual Rudin-Shapiro polynomials of degree n -1 with n = 2 k . In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t) = (1 + η)n has at least An 0.5394282 distinct zeros in [0, 2π) whenever η is real, |η| < 2 -11 , and n is sufficiently large. In this paper we show that the equation R k (t) = (1 + η)n has at least (1/2 -|η| -ε)n/2 distinct zeros in [0, 2π) for every η ∈ (-1/2, 1/2), ε > 0, and sufficiently large k ≥ k η,ε .

arXiv (Cornell University), Sep 20, 2018

We prove that max t∈ [-π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) ... more We prove that max t∈ [-π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) -tan(s/4)) 2n ) for every even trigonometric polynomial Q of degree at most n with complex coefficients satisfying m({t where m(A) denotes the Lebesgue measure of a measurable set A ⊂ R and T 2n is the Chebysev polynomial of degree 2n on [-1, 1] defined by T 2n (cos t) = cos(2nt) for t ∈ R. This inequality is sharp. We also prove that max t∈[-π,π] |Q(t)| ≤ T 2n (sec(s/2)) = 1 2 ((sec(s/2) + tan(s/2)) 2n + (sec(s/2) -tan(s/2)) 2n ) for every trigonometric polynomial Q of degree at most n with complex coefficients satisfying m({t ∈ [-π, π] : |Q(t)| ≤ 1}) ≥ 2π -s , s ∈ (0, π) .

Constructive Approximation, Sep 10, 2013

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for com... more We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein-Szegő-Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new approach to see V.S. Videnskii's Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities published first in 1960. A new Riesz-Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii's Bernstein-type inequality gives Videnskii's Markov-type inequality immediately. Key words and phrases. basic polynomial inequalities, Videnskii's Markov and Bernstein type inequalities for trigonometric polynomials on subintervals of the period, asymptotically sharp Bernstein and Lax type inequalities for complex algebraic polynomials on subarcs of the unit circle, the right Bernstein-Szegő and Riesz-Schur type inequalities for trigonometric polynomials on subintervals of the period..

Mathematische Annalen, Dec 1, 2001

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex pl... more Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := pn : pn(z) = n k=0 1991 Mathematics Subject Classification. 41A17.

arXiv (Cornell University), Dec 31, 1994

Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central t... more Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central tool is a bounded Remez type inequality for non-dense Müntz spaces.

arXiv (Cornell University), Feb 20, 2017

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their... more In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we study the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle. We also show that the Rudin-Shapiro polynomials P k and Q k of degree n -1 with n := 2 k have o(n) zeros on the unit circle. This should be compared with a result of B. Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd primes p the Fekete polynomials f p of degree p -1 have asymptotically κ 0 p zeros on the unit circle, where 0.500813 > κ 0 > 0.500668. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers. We also prove that there are absolute constants c 1 > 0 and c 2 > 0 such that the k-th Rudin-Shapiro polynomials P k and Q k of degree n -1 = 2 k -1 have at least c 2 n zeros in the annulus

Sbornik Mathematics, Apr 30, 2016

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex pl... more Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let P c n denote the set of all algebraic polynomials of degree at most n with complex coefficients. Associated with λ ≥ 0 let The class K 0 n is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (ε n ) of positive numbers tending to 0, we say that a sequence (P n ) of polynomials Although we do not know in general, whether or not ultraflat sequences (P n ) of polynomials P n ∈ K λ n exists, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences (P n ) of conjugate, plain, or skew reciprocal unimodular polynomials 1991 Mathematics Subject Classification. 41A17.

Journal of Approximation Theory, Dec 1, 2003

arXiv (Cornell University), Jun 20, 2012

We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times th... more We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times the Chebyshev constant ) for quite general sets A ⊂ R m with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p > 0, as well as provide an independent proof of their result for p = 4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.

Mathematical proceedings of the Cambridge Philosophical Society, Nov 26, 2008

a j e -(t-λ j ) 2 , a j , λ j ∈ R . We prove that there is an absolute constant c 1 > 0 such that... more a j e -(t-λ j ) 2 , a j , λ j ∈ R . We prove that there is an absolute constant c 1 > 0 such that exp(c 1 (min{n for every s ∈ (0, ∞) and n ≥ 9, where the supremum is taken for all f ∈ G n with This is what we call (an essentially sharp) Remez-type inequality for the class G n . We also prove the right higher dimensional analog of the above result.

Illinois Journal of Mathematics, Dec 1, 1997

Siam Journal on Mathematical Analysis, Mar 1, 1994

We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sh... more We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sharp lower and upper bounds for the corresponding ChristoEefe'. functions, and for the distance of the consecutive zeros of the corresponding orthogonal polynomials. The novelty of our results is that our constants depend only on the degree of the weight function but not on the weight itse!f. !C 1992 Academx Press. Inc.

arXiv (Cornell University), Jan 22, 2020

Polynomials with coefficients in {-1, 1} are called Littlewood polynomials. Using special propert... more Polynomials with coefficients in {-1, 1} are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vallée Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants η 2 > η 1 > 0 and a sequence (P n ) of Littlewood polynomials P n of degree n such that confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence (P n ) of Littlewood polynomials P n is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of P n making the Littlewood polynomials P n close to skew-reciprocal.

Journal D Analyse Mathematique, Nov 1, 2020

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their... more In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we show that the Mahler measure of the Rudin-Shapiro polynomials of degree n -1 with n = 2 k is asymptotically (2n/e) 1/2 , as it was conjectured by B. Saffari in 1985. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers.

Transactions of the American Mathematical Society, Feb 1, 1994

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with res... more The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1 ], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1 ). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp L2 Markov inequality is proved.

arXiv (Cornell University), Apr 26, 2019

We give a simple, elementary, and at least partially new proof of Arestov's famous extension of B... more We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in L p to all p ≥ 0. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic polynomials P n ∈ P c n can be extended to all trigonometric polynomials T n ∈ T c n .

arXiv (Cornell University), Feb 20, 2017

Let n 1 < n 2 < • • • < n N be non-negative integers. In a private communication Brian Conrey ask... more Let n 1 < n 2 < • • • < n N be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials T N (θ) = N j=1 cos(n j θ) tends to ∞ as a function of N . Conrey's question in general does not appear to be easy. Let P n (S) be the set of all algebraic polynomials of degree at most n with each of their coefficients in S. For a finite set S ⊂ C let M = M (S) := max{|z| : z ∈ S}. It has been shown recently that if S ⊂ R is a finite set and (P n ) is a sequence of self-reciprocal polynomials P n ∈ P n (S) with |P n (1)| tending to ∞, then the number of zeros of P n on the unit circle also tends to ∞. In this paper we show that if S ⊂ Z is a finite set, then every self-reciprocal polynomial P ∈ P n (S) has at least c(log log log |P (1)|) 1-ε -1 zeros on the unit circle of C with a constant c > 0 depending only on ε > 0 and M = M (S). Our new result improves the exponent 1/2 -ε in a recent result by Julian Sahasrabudhe to 1 -ε. Sahasrabudhe's new idea is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set S contains only integers is deeply exploited.

arXiv (Cornell University), Oct 9, 2018

Let for every ultraflat sequence (P n ) of polynomials P n ∈ K n and for every q ∈ (0, ∞), where ... more Let for every ultraflat sequence (P n ) of polynomials P n ∈ K n and for every q ∈ (0, ∞), where P * n is the conjugate reciprocal polynomial associated with P n , Γ is the usual gamma function, and the ∼ symbol means that the ratio of the left and right hand sides converges to 1 as n → ∞. Another highlight of the paper states that 1 2π 2π 0 for every ultraflat sequence (P n ) of polynomials P n ∈ K n . We prove a few other new results and reprove some interesting old results as well.

Proceedings of the American Mathematical Society, Apr 7, 2023

Let either R k (t) := |P k (e it)| 2 or R k (t) := |Q k (e it)| 2 , where P k and Q k are the usu... more Let either R k (t) := |P k (e it)| 2 or R k (t) := |Q k (e it)| 2 , where P k and Q k are the usual Rudin-Shapiro polynomials of degree n − 1 with n = 2 k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t) = (1 + η)n has at least An 0.5394282 distinct zeros in [0, 2π) whenever η is real, |η| < 2 −11 , and n is sufficiently large. In this paper we show that the equation R k (t) = (1 + η)n has at least (1/2 − |η| − ε)n/2 distinct zeros in [0, 2π) for every η ∈ (−1/2, 1/2), ε > 0, and sufficiently large k ≥ k η,ε. M q (f) , Key words and phrases. polynomials, restricted coefficients, oscillation of the modulus on the unit circle, Rudin-Shapiro polynomials, oscillation of the modulus on the unit circle, number of real zeros in the period.

arXiv (Cornell University), Feb 7, 2018

, where P k and Q k are the usual Rudin-Shapiro polynomials of degree n -1 with n = 2 k . In a re... more , where P k and Q k are the usual Rudin-Shapiro polynomials of degree n -1 with n = 2 k . In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t) = (1 + η)n has at least An 0.5394282 distinct zeros in [0, 2π) whenever η is real, |η| < 2 -11 , and n is sufficiently large. In this paper we show that the equation R k (t) = (1 + η)n has at least (1/2 -|η| -ε)n/2 distinct zeros in [0, 2π) for every η ∈ (-1/2, 1/2), ε > 0, and sufficiently large k ≥ k η,ε .

arXiv (Cornell University), Sep 20, 2018

We prove that max t∈ [-π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) ... more We prove that max t∈ [-π,π] |Q(t)| ≤ T 2n (sec(s/4)) = 1 2 ((sec(s/4) + tan(s/4)) 2n + (sec(s/4) -tan(s/4)) 2n ) for every even trigonometric polynomial Q of degree at most n with complex coefficients satisfying m({t where m(A) denotes the Lebesgue measure of a measurable set A ⊂ R and T 2n is the Chebysev polynomial of degree 2n on [-1, 1] defined by T 2n (cos t) = cos(2nt) for t ∈ R. This inequality is sharp. We also prove that max t∈[-π,π] |Q(t)| ≤ T 2n (sec(s/2)) = 1 2 ((sec(s/2) + tan(s/2)) 2n + (sec(s/2) -tan(s/2)) 2n ) for every trigonometric polynomial Q of degree at most n with complex coefficients satisfying m({t ∈ [-π, π] : |Q(t)| ≤ 1}) ≥ 2π -s , s ∈ (0, π) .

Constructive Approximation, Sep 10, 2013

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for com... more We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein-Szegő-Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new approach to see V.S. Videnskii's Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities published first in 1960. A new Riesz-Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii's Bernstein-type inequality gives Videnskii's Markov-type inequality immediately. Key words and phrases. basic polynomial inequalities, Videnskii's Markov and Bernstein type inequalities for trigonometric polynomials on subintervals of the period, asymptotically sharp Bernstein and Lax type inequalities for complex algebraic polynomials on subarcs of the unit circle, the right Bernstein-Szegő and Riesz-Schur type inequalities for trigonometric polynomials on subintervals of the period..

Mathematische Annalen, Dec 1, 2001

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex pl... more Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := pn : pn(z) = n k=0 1991 Mathematics Subject Classification. 41A17.

arXiv (Cornell University), Dec 31, 1994

Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central t... more Two relatively long-standing conjectures concerning Müntz polynomials are resolved. The central tool is a bounded Remez type inequality for non-dense Müntz spaces.

arXiv (Cornell University), Feb 20, 2017

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their... more In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we study the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle. We also show that the Rudin-Shapiro polynomials P k and Q k of degree n -1 with n := 2 k have o(n) zeros on the unit circle. This should be compared with a result of B. Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd primes p the Fekete polynomials f p of degree p -1 have asymptotically κ 0 p zeros on the unit circle, where 0.500813 > κ 0 > 0.500668. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers. We also prove that there are absolute constants c 1 > 0 and c 2 > 0 such that the k-th Rudin-Shapiro polynomials P k and Q k of degree n -1 = 2 k -1 have at least c 2 n zeros in the annulus

Sbornik Mathematics, Apr 30, 2016

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex pl... more Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let P c n denote the set of all algebraic polynomials of degree at most n with complex coefficients. Associated with λ ≥ 0 let The class K 0 n is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (ε n ) of positive numbers tending to 0, we say that a sequence (P n ) of polynomials Although we do not know in general, whether or not ultraflat sequences (P n ) of polynomials P n ∈ K λ n exists, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences (P n ) of conjugate, plain, or skew reciprocal unimodular polynomials 1991 Mathematics Subject Classification. 41A17.

Journal of Approximation Theory, Dec 1, 2003

arXiv (Cornell University), Jun 20, 2012

We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times th... more We derive bounds and asymptotics for the maximum Riesz polarization quantity (which is n times the Chebyshev constant ) for quite general sets A ⊂ R m with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when A is the unit circle and p > 0, as well as provide an independent proof of their result for p = 4 that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.

Mathematical proceedings of the Cambridge Philosophical Society, Nov 26, 2008

a j e -(t-λ j ) 2 , a j , λ j ∈ R . We prove that there is an absolute constant c 1 > 0 such that... more a j e -(t-λ j ) 2 , a j , λ j ∈ R . We prove that there is an absolute constant c 1 > 0 such that exp(c 1 (min{n for every s ∈ (0, ∞) and n ≥ 9, where the supremum is taken for all f ∈ G n with This is what we call (an essentially sharp) Remez-type inequality for the class G n . We also prove the right higher dimensional analog of the above result.

Illinois Journal of Mathematics, Dec 1, 1997

Siam Journal on Mathematical Analysis, Mar 1, 1994

We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sh... more We study generalized Jacobi weight functions in terms of their (generalized) degree. We obtain sharp lower and upper bounds for the corresponding ChristoEefe'. functions, and for the distance of the consecutive zeros of the corresponding orthogonal polynomials. The novelty of our results is that our constants depend only on the degree of the weight function but not on the weight itse!f. !C 1992 Academx Press. Inc.

arXiv (Cornell University), Jan 22, 2020

Polynomials with coefficients in {-1, 1} are called Littlewood polynomials. Using special propert... more Polynomials with coefficients in {-1, 1} are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vallée Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants η 2 > η 1 > 0 and a sequence (P n ) of Littlewood polynomials P n of degree n such that confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence (P n ) of Littlewood polynomials P n is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of P n making the Littlewood polynomials P n close to skew-reciprocal.

Journal D Analyse Mathematique, Nov 1, 2020

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their... more In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we show that the Mahler measure of the Rudin-Shapiro polynomials of degree n -1 with n = 2 k is asymptotically (2n/e) 1/2 , as it was conjectured by B. Saffari in 1985. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers.

Transactions of the American Mathematical Society, Feb 1, 1994

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with res... more The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1 ], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1 ). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp L2 Markov inequality is proved.

arXiv (Cornell University), Apr 26, 2019

We give a simple, elementary, and at least partially new proof of Arestov's famous extension of B... more We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in L p to all p ≥ 0. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic polynomials P n ∈ P c n can be extended to all trigonometric polynomials T n ∈ T c n .

arXiv (Cornell University), Feb 20, 2017

Let n 1 < n 2 < • • • < n N be non-negative integers. In a private communication Brian Conrey ask... more Let n 1 < n 2 < • • • < n N be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials T N (θ) = N j=1 cos(n j θ) tends to ∞ as a function of N . Conrey's question in general does not appear to be easy. Let P n (S) be the set of all algebraic polynomials of degree at most n with each of their coefficients in S. For a finite set S ⊂ C let M = M (S) := max{|z| : z ∈ S}. It has been shown recently that if S ⊂ R is a finite set and (P n ) is a sequence of self-reciprocal polynomials P n ∈ P n (S) with |P n (1)| tending to ∞, then the number of zeros of P n on the unit circle also tends to ∞. In this paper we show that if S ⊂ Z is a finite set, then every self-reciprocal polynomial P ∈ P n (S) has at least c(log log log |P (1)|) 1-ε -1 zeros on the unit circle of C with a constant c > 0 depending only on ε > 0 and M = M (S). Our new result improves the exponent 1/2 -ε in a recent result by Julian Sahasrabudhe to 1 -ε. Sahasrabudhe's new idea is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set S contains only integers is deeply exploited.

arXiv (Cornell University), Oct 9, 2018

Let for every ultraflat sequence (P n ) of polynomials P n ∈ K n and for every q ∈ (0, ∞), where ... more Let for every ultraflat sequence (P n ) of polynomials P n ∈ K n and for every q ∈ (0, ∞), where P * n is the conjugate reciprocal polynomial associated with P n , Γ is the usual gamma function, and the ∼ symbol means that the ratio of the left and right hand sides converges to 1 as n → ∞. Another highlight of the paper states that 1 2π 2π 0 for every ultraflat sequence (P n ) of polynomials P n ∈ K n . We prove a few other new results and reprove some interesting old results as well.

Proceedings of the American Mathematical Society, Apr 7, 2023

Let either R k (t) := |P k (e it)| 2 or R k (t) := |Q k (e it)| 2 , where P k and Q k are the usu... more Let either R k (t) := |P k (e it)| 2 or R k (t) := |Q k (e it)| 2 , where P k and Q k are the usual Rudin-Shapiro polynomials of degree n − 1 with n = 2 k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t) = (1 + η)n has at least An 0.5394282 distinct zeros in [0, 2π) whenever η is real, |η| < 2 −11 , and n is sufficiently large. In this paper we show that the equation R k (t) = (1 + η)n has at least (1/2 − |η| − ε)n/2 distinct zeros in [0, 2π) for every η ∈ (−1/2, 1/2), ε > 0, and sufficiently large k ≥ k η,ε. M q (f) , Key words and phrases. polynomials, restricted coefficients, oscillation of the modulus on the unit circle, Rudin-Shapiro polynomials, oscillation of the modulus on the unit circle, number of real zeros in the period.