The t-core of an s-core

We consider the t-core of an s-core partition, when s and t are coprime positive integers. Olsson has shown that the t-core of an s-core is again an s-core, and we examine certain actions of the affine symmetric group on s-cores which preserve the t-core of an s-core. Along the way, we give a new proof of Olsson?s result. We also give a new proof of a result of Vandehey, showing that there is a simultaneous s- and t-core which contains all others.     

An asymptotic formula for the t-core partition function and a conjecture of Stanton

For a positive integer t, a partition is said to be a t-core if each of the hook numbers from its Ferrers-Young diagram is not a multiple of t. In 1996, Granville and Ono proved the t-core partition conjecture, that at(n), the number of t-core partitions of n, is positive for every nonnegative integer n as long as t?4. As part of their proof, they showed that if p?5 is prime, the generating function for ap(n) is essentially a multiple of an explicit Eisenstein Series together with a cusp form. This representation of the generating function leads to an asymptotic formula for ap(n) involving L-functions and divisor functions. In 1999, Stanton conjectured that for t?4 and n?t+1, at(n)?at+1(n). Here we prove a weaker form of this conjecture, that for t?4 and n sufficiently large, at(n)?at+1(n). Along the way, we obtain an asymptotic formula for at(n) which, in the cases where t is coprime to 6, is a generalization of the formula which follows from the work of Granville and Ono when t=p?5 is prime.     

Congruences for t-core partition functions

Let t?2$t?2$ be an integer and p?5$p?5$ be a prime. We prove a conjecture on congruences for 2^t$2^t$-core partition functions. We also find many new congruences for p  -core partition functions when 5?p?47$5?p?47$.     

Explicit constructions of infinite families of MSTD sets

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We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A|>|A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r⁴ of the r² subsets of {1,…,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/²r/2 for some polynomial f(r)). We conclude by generalizing our method to compare linear forms ?₁A+?+?nA with ?i∈{−1,1}.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=vIDDa1R2.     

Minimal zero sum sequences of length four over finite cyclic groups

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Let G be a finite cyclic group. Every sequence S over G can be written in the form S=(n₁g)⋅…⋅(nlg) where g∈G and n₁,…,nl∈[1,ord(g)], and the index ind(S) of S is defined to be the minimum of (n₁+?+nl)/ord(g) over all possible g∈G such that 〈g〉=〈supp(S)〉. The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd(|G|,6)=1, then every minimal zero-sum sequence of length 4 has index 1.

Video

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On the behaviour of p-adic L-functions

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Let Lp(s,χ) denote a Leopoldt-Kubota p-adic L-function, where p>2 and χ is a nonprincipal even character of the first kind. The aim of this article is to study how the values assumed by this function depend on the Iwasawa λ-invariant associated to χ. Assuming that λ?p−1, it turns out that Lp(s,χ) behaves, in some sense, like a polynomial of degree λ. The results lead to congruences of a new type for (generalized) Bernoulli numbers.

Video

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Note on integer powers in sumsets

Let k,m,n?2 be integers. Let A be a subset of {0,1,…,n} with 0∈A and the greatest common divisor of all elements of A is 1. Suppose that

     

Distinct length modular zero-sum subsequences: A proof of Graham's conjecture

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Let S be a sequence of n nonnegative integers not exceeding n−1 such that S takes at least three distinct values. We show that S has two nonempty zero-sum subsequences with distinct lengths. This proves a conjecture of R.L. Graham. The validity of this conjecture was verified by Erd?s and Szemerédi for all sufficiently large prime n.

Video

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Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

Euler numbers congruences and Dirichlet L-functions

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In this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275-289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values L(s,χ) at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs.     

A new result on Davenport constant

Let G be a finite abelian group of order n and Davenport constant D(G). Let S=⁰h(S)_∏g∈Ggvg(S)∈F(G) be a sequence with a maximal multiplicity h(S) attained by 0 and t=|S|?n+D(G)−1. Then 0∈k_∑(S) for every 1?k?t+1−D(G). This is a refinement of the fundamental result of Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103].     

Picard sequences with every orbit containing infinitely many perfect n-powers

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Finding a function which generates a sequence via iteration whose values at one or many points in its domain satisfy certain prescribed properties, i.e., finding a function such that the Picard orbit(s) of one or many points in its domain which possess some given properties, is an interesting problem. Given any positive integer n greater than one, we construct in this paper families of functions on the natural numbers such that the sequence of the iterations of each of these functions at any positive integer s contains infinitely many perfect n-powers. In terms of Picard sequences, this amounts to constructing a function whose Picard orbit at every point in its domain contains infinitely many perfect n-powers.

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A covering system whose smallest modulus is 40

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Paul Erd?s, in 1950, asked whether for each positive integer N there exists a finite set of congruence classes, with distinct moduli, covering the integers, whose smallest modulus is N. In this vein, we construct a covering system of the integers with smallest modulus N=40.

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Constructing MSTD sets using bidirectional ballot sequences

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We construct a new dense family of MSTD subsets of {0,1,2,…,n−1}. Our construction gives Θ(n²/n) MSTD sets, improving the previous best construction with Ω(n²/n⁴) MSTD sets by Miller, Orosz, and Scheinerman.     

An application of Fourier transforms on finite Abelian groups to an enumeration arising from the Josephus problem

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We analyze an enumeration associated with the Josephus problem by applying a Fourier transform to a multivariate generating function. This yields a formula for the enumeration that reduces to a simple expression under a condition we call local prime abundance. Under this widely held condition, we prove (Corollary 3.4) that the proportion of Josephus permutations in the symmetric group Sn that map t to k (independent of the choice of t and k) is 1/n. Local prime abundance is intimately connected with a well-known result of S.S. Pillai, which we exploit for the purpose of determining when it holds and when it fails to hold. We pursue the first case where it fails, reducing an intractable DFT computation of the enumeration to a tractable one. A resulting computation shows that the enumeration is nontrivial for this case.

Video

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Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.     

On the zeta function associated with module classes of a number field

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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.

Video

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A semialgebraic closure for commutative algebra

Let A⊂R be rings containing the rationals. In R let S be a multiplicatively closed subset such that 1∈S and 0∉S, T a preorder of R (a proper subsemiring containing the squares) such that S⊂T and I an A-submodule of R. Define ρ(I) (or ρ_S,T(I)) to be

ρ(I)={a∈R|sa^2m+t∈I^2m for some m∈N,s∈S and t∈T}.     

On the Erd?s-Ginzburg-Ziv constant of finite abelian groups of high rank

Let G be a finite abelian group. The Erd?s-Ginzburg-Ziv constant s(G) of G is defined as the smallest integer l∈N such that every sequence S over G of length |S|?l has a zero-sum subsequence T of length |T|=exp(G). If G has rank at most two, then the precise value of s(G) is known (for cyclic groups this is the theorem of Erd?s-Ginzburg-Ziv). Only very little is known for groups of higher rank. In the present paper, we focus on groups of the form , with n,r∈N and n?2, and we tackle the study of s(G) with a new approach, combining the direct problem with the associated inverse problem.     

Simple arguments on consecutive power residues

By some extremely simple arguments, we point out the following:

(i)

If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.

(ii)

Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .