Abstract—The maximum known asymptotic merit factor for binary sequences has been stuck at a value of 6 since the 1980s. Several authors have suggested that this value cannot be improved. In this paper, we construct an infinite family of binary sequences whose asymptotic merit factor we conjecture to be greater than 6 34. We present what we believe to be compelling evidence in support of this conjecture. The numerical experimentation that led to this construction is a significant part of the story. Index Terms—Aperiodic autocorrelation, asymptotic, binary sequence, merit factor. I. PREAMBLE WE begin with a general discussion of the merit factor problem for binary sequences and outline our approach, prior to a more formal introduction in Section II. The problem of determining the maximal merit factor for binary sequences

Abstract. A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.

this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4], [31], [43], [16], etc.) In the classical case E = [-1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) := 2 1-n cos(n arccos x), n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so that t n (x) := # b - a 2 # n T n # 2x - a - b b - a # is a monic polynomial with real coe#cients and the smallest uniform norm on [a, b] among all monic polynomials from P n (C). In fact, (1.1) #t n # [a,b] = 2 # b - a 4 # n , n # N, and we find that the Chebyshev constant for [a, b] is given by (1.2) t C ([a, b]) := lim n## #t n # 1/n [a,b] = b - a 4 . The Chebyshev constant of an arbitrary compact set E # C is defined in a similar fashion: (1.3) t C (E) := lim n## #t n # 1/n E , where t n is the Chebyshev polynomial of degree n on E. It is known that t C (E) is equal to the transfinite diameter and the logarithmic capacity cap(E) of the set E (cf. [43, pp. 71-75], [16] and [30] for the definitions and background material). 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter, zeros, multiple factors, asymptotic...

Abstract. In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in C give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of Z[X] of degree at most 36 with Mahler measure less than 1.324... and of degree 38 and 40 with Mahler measure less than 1. 31. 1.

the nm are integral and all different, what is the lower bound on the number of real zeros of P N m=1 cos(nmθ)? Possibly N − 1, or not much less. ” Here real zeros are counted in a period. In fact no progress appears to have been made on this in the last half century. In a recent paper [2] we showed that this is false. There exists a cosine polynomial P N m=1 cos(nmθ) with the nj integral and all different so that the number of its real zeros in the period is O(N 9/10 (log N) 1/5) (here the frequencies nm = nm(N) may vary with N). However, there are reasons to believe that a cosine polynomial P N m=1 cos(nmθ) always has many zeros on the period. Denote the number of zeros of a trigonometric polynomial T in the period [−π, π) by N (T). In this paper we prove the following. Theorem. Suppose the set {aj: j ∈ N} ⊂ R is finite and the set {j ∈ N: aj � = 0} is infinite. Let nX Tn(t) = aj cos(jt). Then limn→ ∞ N (Tn) = ∞. j=0 One of our main tools, not surprisingly, is the resolution of the Littlewood Conjecture [4]. 1.

Abstract. We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter tM(I) is defined as the infimum of all such supremums. We show that if I has length 1 then tM(I) = 1 2. We make three general conjectures relating to the value of tM(I) for intervals I of length less that 4. We also conjecture a value for tM([0, b]) where 0 < b ≤ 1. We give some partial results, as well as computational evidence, to support these conjectures. We define functions L−(t) and L+(t), which measure properties of the lengths of intervals I with tM(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining tM(I) when I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

Abstract. In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that log n δn: = 33π √ n satisfies δn ≤ 1. We show that any polynomial in

The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.

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We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = ∑ n≥1 snβ−n, with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.