On additive representation function of general sequences

Let 0 ≦ a ₁ < a ₂ < ? be an infinite sequence of integers and let r ₁(A, n) = |(i;j): a _i + a _j = n, i ≦ j|. We show that if d > 0 is an integer, then there does not exist n ₀ such that d ≦ r ₁ (A, n) ≦ d + [√2d + ½] for n > n ₀.     

On Tverberg partitions

A theorem of Tverberg from 1966 asserts that every set X ? ? ^d of n = T(d, r) = (d + 1)(r ? 1) + 1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a ₁,..., a _r satisfying n = a ₁ + ··· + a _r ), in which the parts a _i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a _i ≤ d + 1 for all i = 1,..., r, there exists a set X ? ? of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a ₁,..., a _r .     

On the modular sumset partition problem

A sequence m₁≥m₂≥?≥m_k of k positive integers isn-realizable if there is a partition X₁,X₂,…,X_k of the integer interval [1,n] such that the sum of the elements in X_i is m_i for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)³, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.     

A note on lattice chains and Delannoy numbers

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0?a_i?n_i for 1?i?d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number of chains in L. Then we derive a second, alternative, expression for the total number of chains in L when d=2. Setting n₁=n₂ in this expression yields a new proof of a result of Stanley [Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999] relating the total number of chains to the central Delannoy numbers. We also conjecture a generalization of Stanley's result to higher dimensions.     

On a partition problem of Frobenius

We consider the following problem, which was raised by Frobenius: Given n relatively prime positive integers, what is the largest integer M(a₁, a₂, …, a_n) omitted by the linear form Σ_i=1ⁿa_ix_i, where the x_i are variable nonnegative integers. We give the solution for certain special cases when n = 3.     

Irreducibility of Hurwitz spaces of coverings with one special fiber

Let Y be a smooth, projective complex curve of genus g ? 1. Let d be an integer ? 3, let e = {e₁, e₂,..., e_r} be a partition of d and let |e| = Σ_i=1^r(e_i − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group S_d. We prove the irreducibility of these Hurwitz spaces when n − 1 + |e| ? 2d, thus generalizing a result of Graber, Harris and Starr [A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056].     

Number of Zeros of Diagonal Polynomials over Finite Fields

Let F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+···+cnxdnn=c over the finite fields, where d1∣q−1, ciϵF*(i=1, 2,…,n), and cϵF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1∣N, where ri is the least integer such that di∣pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb∣N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n.     

A note on the triangle conjecture

We prove some particular cases of the following conjecture of Perrin and Schützenberger, known as “the triangle conjecture.” Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {aⁱba^j| 0 ? i + j ? d}. If X ? B_d is a code, then |X| ? d + 1.     

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSp _r × GSp_2m to GSp _r+2m for some positive integers r and m.     

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if , where is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2^k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φ_p^k(4) are universally bad, where p is prime and φ_n(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φ_p^k(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ_2^k(4), and more generally, those of the form 4^k+1, are not de Bruijn universally bad.     

Extremal values of continuants and transcendence of certain continued fractions

We prove a criterion for the transcendence of continued fractions whose partial quotients are contained in a finite set {b₁,…,b_r} of positive integers such that the density of occurrences of b_i in the sequence of partial quotients exists for 1ir. As an application we study continued fractions [0,a₁,a₂,a₃,…] with a_n=1+([nθ]modd) where θ is irrational and d2 is a positive integer.     

Representations of integers by linear forms in nonnegative integers

Let Ω be the set of positive integers that are omitted values of the form f = Σ_i=1ⁿa_ix_i, where the a_i are fixed and relatively prime natural numbers and the x_i are variable nonnegative integers. Set ω = #Ω and κ = max Ω + 1 (the conductor). Properties of ω and κ are studied, such as an estimate for ω (similar to one found by Brauer) and the inequality 2ω ≥ κ. The so-called Gorenstein condition is shown to be equivalent to 2ω = κ.     

On discount residue classes

It is shown that, whenever m₁, m₂,…, m_n are natural numbers such that the pairwise greatest common divisors, d_ij=(m_i, m_j), i≠j are distinct and different from 1, then there exist integers a₁, a₂,…,a_n such that the solution sets of the congruences x≡_i (modm_i), i= 1,2,…,n are disjoint.     

On exact coverings of the integers

By an exact covering of modulusm, we mean a finite set of liner congruencesx≡a _i (modm _i ), (i=1,2,...r) with the properties: (I)m _i ∣m, (i=1,2,...,r); (II) Each integer satisfies precisely one of the congruences. Let α≥0, β≥0, be integers and letp andq be primes. Let μ (m) senote the Möbius function. Letm=p ^α q ^β and letT(m) be the number of exact coverings of modulusm. Then,T(m) is given recursively by $$\mathop \Sigma \limits_{d/m} \mu (d)\left( {T\left( {\frac{m}{d}} \right)} \right)^d = 1$$ .     

On the nonvanishing of homogeneous product sums

For integer n ≥ 1 let H_n = H_n(x, y, z) = Σ_{p + q + r = n}x^py^qz^r be the homogeneous product sum of weight n on three letters x, y, z. Morgan Ward conjectured that H_n ≠ 0 for all integers n, x, y, z with n > 1 and xyz ≠ 0. In support of this conjecture he proved that H_n ≠ 0 if n is even or if n + 2 is a prime number greater than 3. This paper adds considerably more evidence in support of Ward's conjecture by showing that in many cases H_n(a, b, c)¬=0 modulo 2, 4, or 16. The parity of H_n(a, b, c) is determined in all cases and, when H_n(a, b, c) is even, further congruences are given modulo 4 or 16.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

Langford sequences: perfect and hooked

A sequence {d, d+1,…, d+m?1} of m consecutive positive integers is said to be perfect if the integers {1, 2,…, 2m} can be arranged in disjoint pairs {(a_i, b_i): 1?i?m} so that {b_i?a_i: 1?i?m}={d,d+1,…,d+m?1}. A sequence is hooked if the set {1, 2,…, 2m?1 2m+1} can be arranged in pairs to satisfy the same condition. Well known necessary conditions for perfect sequences are herein shown to be sufficient. Similar necessary and sufficient conditions for hooked sequences are given.     

A linear algorithm for nonhomogeneous spectra of numbers

Given a sequence of integers [a_i]_i=1ⁿ, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that a_i = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of $\text{d}$ and $\text{d}$ such that $\text{d < α <}\text{d}$ if and only if a_i = [iα + β] (1 ? i ? n) for suitable β = β(α).     

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

On the density of odd integers of the form (p − 1)2−n and related questions

It is shown that odd integers k such that k · 2ⁿ + 1 is prime for some positive integer n have a positive lower density. More generally, for any primes p₁, …, p_r, the integers k such that k is relatively prime to each of p₁,…, p_r, and such that k · p₁^n₁p₂^n₂ … p_r^n_r + 1 is prime for some n₁,…, n_r, also have a positive lower density.