Showing papers in "Acta Arithmetica in 2006"
TL;DR: In this paper, the power of prefixes of Sturmian sequences was investigated in terms of its S-adic representation and an explicit formula for ice(ω), the initial critical exponent, was given, defined as the supremum of all real numbers p > 0.
Abstract: In this paper we investigate powers of prefixes of Sturmian sequences. We give an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p > 0 for which there exist arbitrary long prefixes of ω of the form up, in terms of its S-adic representation. This formula is based on Ostrowski's numeration system. Furthermore we characterize those irrational slopes α of which there exists a Sturmian sequence ω beginning in only finitely many powers of 2 + e, that is for which ice(ω) = 2. In the process we recover the known results for the index (or critical exponent) of a Sturmian sequence. We also focus on the Fibonacci Sturmian shift and prove that the set of Sturmian sequences with ice strictly smaller than its everywhere value has Hausdorff dimension 1.
90 citations
TL;DR: Ahlgren and Ono as mentioned in this paper proved that the a(p) for odd primes p is related to the modular Calabi-Yau threefold (1.4) problem.
Abstract: (modp). This was proved for primes p such that p ∤ a(p) by Ishikawa [8] and unconditionally by Ahlgren and Ono [1]. The a(p) for odd primes p are also known to be related to the modular Calabi–Yau threefold (1.4) x + 1 x + y + 1 y + z + 1 z + w + 1 w = 0, in the following way. Let N(p) denote the number of solutions to (1.4) over the finite field with p elements. Then Ahlgren and Ono [2], van Geemen and
79 citations
66 citations
TL;DR: In this article, a family of dynamical systems which are related to generalized radix representations are studied and the properties of these systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.
Abstract: We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.
59 citations
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TL;DR: The case of t = 0, that is T (s, 0, u), is called the Euler-Zagier double zeta function [10] as mentioned in this paper.
Abstract: The case of t = 0, that is T (s, 0, u), is called the Euler–Zagier double zeta function [10]. The values T (a, b, c) for a, b, c ∈ N were first investigated by Tornheim [7] in 1950 and later Mordell [5] in 1958. Tornheim [7, Theorem 7] showed that T (a, b, c) can be expressed as a polynomial in {ζ(j) | 2 ≤ j ≤ a+ b+ c} with rational coefficients when a + b + c is odd, and that the same is true for T (2r, 2r, 2r) and T (2r− 1, 2r, 2r+1) [7, Theorem 8], but he did not give the coefficients. Mordell [5, Theorem III] proved that T (2r, 2r, 2r) = krπ 6r for some rational number kr. In 1985 Subbarao and Sitaramachandrarao [6, Theorem 4.1] explicitly determined T (2p, 2q, 2r)+T (2q, 2r, 2p)+T (2r, 2p, 2q) (p, q, r ∈ N). Then, by taking p = q = r, they gave an explicit formula for T (2r, 2r, 2r) (r ∈ N) [6, Remark 3.1]. In 1996 Huard, Williams and Zhang [3, Theorems 1–3] determined T (r, 0, N−r) (r ∈ N, N ∈ 2N+1, 1 ≤ r ≤ N−2), T (p, q,N − p − q) (p, q ∈ N ∪ {0}, N ∈ 2N + 1, 1 ≤ p + q ≤ N − 1, 0 ≤ p, q ≤ N − 2) and T (r, r, r) (r ∈ N). In 2002 Tsumura [8, Theorem 1]
43 citations
TL;DR: A systematic study of the genera and plane models of quotient curves of the Suzuki curve y 2s+1 − y = x 2s(x 22s + 1 − x), s ∈ N, is carried out in this paper.
Abstract: A systematic study of the genera and plane models of quotient curves of the Suzuki curve y 2s+1 − y = x2s(x22s+1 − x), s ∈ N, is carried out.
40 citations
38 citations
TL;DR: In this article, the sequence of best approximations for non totally Pisot numbers which satisfy a combinatorial property called property (F) was computed, and the results highly depend on topological properties of the Rauzy fractal associated to the numeration system induced by the Pisot number.
Abstract: In this paper, we compute the sequence of best approximations for non totally Pisot numbers which satisfy a combinatorial property called property (F). Our results highly depend on topological properties of the Rauzy fractal associated to the numeration system induced by the Pisot number.
TL;DR: In this article, the truncated determinants of truncated Toeplitz matrices were obtained for the case where f is multiplicative, linking them to the determinant of a truncated TOEPLITZ matrix.
Abstract: In this paper we study matrices A = (aij) whose (i;j) th -entry is a function of i=j; that is, aij = f(i=j) for some f :Q + !C. We obtain a formula for the truncated determinants in the case where f is multiplicative, linking them to determinants of truncated Toeplitz matrices. We apply our formula to obtain several determinants of number-theoretic matrices.
TL;DR: In this article, the authors remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
Abstract: We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
TL;DR: Theorem 1.1.1 as discussed by the authors deals with the most common types of such averages, where the outer summation is over some family of characters, possibly to various moduli.
Abstract: where Λ(n) is the von Mangoldt function and the outer summation is over some family of characters, possibly to various moduli. Our main result is Theorem 1.1 below, which deals with the most common types of such averages. Let m ≥ 1, r ≥ 1, and Q ≥ r. We consider a set H(m, r,Q) of characters χ = ξψ modulo mq, where ξ is a character modulo m and ψ is a primitive character modulo q, with r ≤ q ≤ Q, r | q, and (q,m) = 1. Our result is as follows. Theorem 1.1. Let m ≥ 1, r ≥ 1, Q ≥ r, T ≥ 2, N ≥ 2, and H(m, r,Q) be a set of characters as described above. Then
TL;DR: For Bernoulli and Euler polynomials, the authors showed that the symmetric relation implies the existence of symmetric identities of Miki and Matiyasevich.
Abstract: We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then we have $$rF(s,t;x,y)+sF(t,r;y,z)+tF(r,s;z,x)=0$$ where $$F(s,t;x,y):=\sum_{k=0}^n(-1)^k\binom{s}{k}\binom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as $$\sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0 k
ot=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.$$
TL;DR: In this paper, it was proved that two Pell equations have at most two solutions in positive integers, which is the best possible result, since there are examples of pairs of Pell equations having two positive solutions.
Abstract: It is proved that two Pell equations have at most two solutions in positive integers. This is the best possible result, since there are examples of pairs of Pell equations having two positive solutions.
TL;DR: The Petersson trace formula as discussed by the authors relates spectral data coming from cusp forms to Kloosterman sums and Bessel functions and can be regarded as the first type of trace formula for automorphic forms.
Abstract: The Petersson trace formula relates spectral data coming from cusp forms to Kloosterman sums and Bessel functions. It was discovered in 1932 [Pe] long before Selberg’s trace formula and can be regarded as the first type of trace formula for automorphic forms. It has proven to be an indispensable tool for estimating the size of the Fourier coefficients of modular forms in many situations. See for example [Se], [IK], and Section 5 of [Iw]. In this paper we will use the relative trace formula to prove a variant of the Petersson trace formula. The resulting generalized formula relates Hecke eigenvalues, Fourier coefficients and Petersson norms of cusp forms (on the spectral side) to Bessel functions and Kloosterman sums (on the geometric side). To state this result, let Sk(N,ω ′) be the space of cusp forms of level N , weight k > 2, and nebentypus ω′ (see Section 3). For an integer n which is prime to N , let F be an orthogonal basis consisting of eigenfunctions for the Hecke operator Tn. Then (see Theorem 3.9)
TL;DR: In this paper, it was shown that a large family of algebraic elements of degree 4 over K are hyperquadratic, i.e. they satisfy α = (Aαr +B)/(Cαr+D), where r is a power of p ≥ 3.
Abstract: Let K be a field of char. p ≥ 3. We prove that a large family of algebraic elements of degree 4 over K are hyperquadratic, i.e. they satisfy α = (Aαr +B)/(Cαr+D), where r is a power of p. In the case where K = Fq(T ), hyperquadratic power series over Fq are analogues of quadratic real numbers. This work is connected with diophantine approximation and continued fractions in fields of power series.
TL;DR: In this paper, the average order of a suitable arithmetic function h is investigated as it ranges over the values taken by an integral binary form F and a general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit.
Abstract: Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit.
TL;DR: The Sato-Tate conjecture was shown to be true on average in a more general setting in this paper, where the density of the Frobenius elements is given by 2 π R β α √ 1− t2 dt.
Abstract: The Sato-Tate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) have density given by 2 π R β α √ 1− t2 dt. We prove that this conjecture is true on average in a more general setting.
TL;DR: In this paper, the density of integers that may be written as p + 2, where p is a prime and k is a nonnegative integer, was investigated, and the density was shown to be polynomial.
Abstract: We investigate the density of integers that may be written as p + 2, where p is a prime and k a nonnegative integer.
TL;DR: In this article, a combinatorial approach was used to obtain a nontrivial lower bound of the p-adic order of the sum X k r (mod p ) n k (1) k f k r p, where p is a prime and f(x) is an integer-valued polyno-mial.
Abstract: Let p be a prime, and let f(x) be an integer-valued polyno- mial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum X k r (mod p ) n k ( 1) k f k r p ,
TL;DR: In this article, the problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)-equations (again, besides certain trivial solutions) was studied.
Abstract: A linear equation on k unknowns is called a (k, h)-equation if it is of the form ∑k i=1 aixi = 0, with ai ∈ {−h, . . . , h} and ∑ ai = 0. For a (k, h)-equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arithmetic progressions, are some of the most well studied problems in additive number theory. Ruzsa was the first to address the general problem of the influence of certain properties of equations on rE(n). His results suggest, but do not imply, that for every fixed k, all but an O(1/h) fraction of the (k, h)-equations E are such that rE(n) > n1−o(1). In this paper we address the generalized problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)-equations (again, besides certain trivial solutions). We denote this quantity by rS(n). Our main result is that all but an O(1/h) fraction of the sets of (k, h)-equations S of size k − b √ 2kc + 1, are such that rS(n) > n1−o(1). We also give several additional results relating properties of sets of equations and rS(n).
TL;DR: In this article, the authors extend the well-known theorem of Vosper and a recent result of Rodseth and one of the present authors to include A and B in arithmetic progressions with the same difference and of size |A+2 and |B|+2 respectively.
Abstract: Let A and B be subsets of Z/pZ such that |A+B| 3, |B|>4, |A+B| 52, then A and B are included in arithmetic progressions with the same difference and of size |A|+2 and |B|+2 respectively. This extends the well-known theorem of Vosper and a recent result of Rodseth and one of the present authors.
TL;DR: In this article, the authors considered numeration schemes defined in terms of dynamical systems and studied the set of reals which obey some constraints on their digits, and showed that almost all sets have zero Lebesgue measure.
Abstract: This paper considers numeration schemes, defined in terms of dynamical systems and studies the set of reals which obey some constraints on their digits. In this general setting, (almost) all sets have zero Lebesgue measure, even though the nature of the constraints and the numeration schemes are very different. Sets of zero measure appear in many areas of science, and Hausdorff dimension has shown to be an appropriate tool for studying their nature.
TL;DR: The Erdős-Ginzburg-Ziv Theorem as mentioned in this paper is a generalization of the pigeonhole principle for m pigeons and 2 boxes, which states that often the best way to avoid zero-sums is to consider sequences with very few distinct terms.
Abstract: A sequence S of terms from an abelian group is zero-sum if the sum of the terms of S is zero. In 1961 Erdős, Ginzburg and Ziv proved that any sequence of 2m− 1 terms from an abelian group of order m contains an m-term zero-sum subsequence [10]. This sparked a flurry of generalizations, variations and extensions [1] [3] [7] [8] [11] [13] [14] [15] [16] [17] [18] [22] [26] [27] [28] [37]. Since a sequence from the cyclic group Z/mZ consisting of only 0’s and 1’s has its m-term zerosum subsequences in exact correspondence with its m-term monochromatic subsequences, then the Erdős-Ginzburg-Ziv Theorem can be viewed as a generalization of the pigeonhole principle for m pigeons and 2 boxes. In essence, the Erdős-Ginzburg-Ziv Theorem expresses the idea that often the best way to avoid zero-sums is to consider sequences with very few distinct terms. For sequences whose length is greater than 2m−1, a natural question to ask is how many m-term zero-sum subsequences can one expect. If the sequence S has length n and consists of at most two distinct terms, then there will be at least (dn 2 e m ) + (bn 2 c m ) m-term monochromatic subsequences. Thus ∗AMS Subject Classification: 11B75 (11B50)