Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q) ⁿ , n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3) ⁿ || < 2/5, ||ζ (5/3) ⁿ || > 1/10 and ||τ (3/2)²ⁿ || < 14/45 hold for each integer ${n \geqslant 0}$ .     

On the powers of 3/2 and other rational numbers

Letp > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ (p /q)ⁿ }, n = 0, 1, 2, …, for certain non‐zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ (3/2)ⁿ }, n ∈ N . The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ (3/2)ⁿ } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the distance from a rational power to the nearest integer

We prove that for any non-zero real number ξ the sequence of fractional parts {ξ(3/2)ⁿ}, n=1,2,3,…, contains at least one limit point in the interval [0.238117…,0.761882…] of length 0.523764…. More generally, it is shown that every sequence of distances to the nearest integer ||ξ(p/q)ⁿ||, n=1,2,3,…, where p/q>1 is a rational number, has both ‘large’ and ‘small’ limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue-Morse sequence and, for irrational ξ, are best possible for every pair p>1, q=1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence ||ξαⁿ||, n=1,2,3,…, where α>1 is an algebraic number and where ξ≠0 is an arbitrary real number satisfying ξ∉Q(α) in case α is a Pisot or a Salem number.     

Adams theorem on Bernoulli numbers revisited

If we denote B_n to be nth Bernoulli number, then the classical result of Adams (J. Reine Angew. Math. 85 (1878) 269) says that p^?|n and (p−1)?n, then p^?|B_n where p is any odd prime p>3. We conjecture that if (p−1)?n, p^?|n and p?+1?n for any odd prime p>3, then the exact power of p dividing B_n is either ? or ?+1. The main purpose of this article is to prove that this conjecture is equivalent to two other unproven hypotheses involving Bernoulli numbers and to provide a positive answer to this conjecture for infinitely many n.     

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let Un(Fq) denote the group of unipotent n×n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of Un+1(Fq) are indexed by lattice paths from the origin to the line x+y=n using the steps (1,0), (1,1), (0,1), (0,2), which are labeled in a certain way by nonzero elements of Fq. In particular, we prove for n?1 that the number of Heisenberg characters of Un+1(Fq) is a polynomial in q−1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of Un(Fq) is a polynomial in q−1 whose coefficients are Delannoy numbers and whose values give a q-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of Un(Fq) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q−1 with nonnegative integer coefficients.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Path-fan Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers R(P_n,F_m), where P_n is a path on n vertices and F_m is the graph obtained from m disjoint triangles by identifying precisely one vertex of every triangle (F_m is the join of K₁ and mK₂). We determine the exact values of R(P_n,F_m) for the following values of n and m: 1?n?5 and m?2; n?6 and 2?m?(n+1)/2; 6?n?7 and m?n-1; n?8 and n-1?m?n or ((q·n-2q+1)/2?m?(q·n-q+2)/2 with 3?q?n-5) or m?(n-3)²/2; odd n?9 and ((q·n-3q+1)/2?m?(q·n-2q)/2 with 3?q?(n-3)/2) or ((q·n-q-n+4)/2?m?(q·n-2q)/2 with (n-1)/2?q?n-5). Moreover, we give nontrivial lower bounds and upper bounds for R(P_n,F_m) for the other values of m and n.     

Path-kipas Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers , where P_n is a path on n vertices and is the graph obtained from the join of K₁ and P_m. We determine the exact values of for the following values of n and m: 1?n?5 and m?3; n?6 and (m is odd, 3?m?2n-1) or (m is even, 4?m?n+1); 6?n≤7 and m=2n-2 or m?2n; n?8 and m=2n-2 or m=2n or (q·n-2q+1?m?q·n-q+2 with 3?q?n-5) or m?(n-3)²; odd n?9 and (q·n-3q+1?m?q·n-2q with 3?q?(n-3)/2) or (q·n-q-n+4?m?q·n-2q with (n-1)/2?q?n-4). Moreover, we give lower bounds and upper bounds for for the other values of m and n.     

Exact number of positive solutions for a class of semipositone problems

This paper is concerned with the boundary value problems y″+λ(y^p−y^q)=0 and y(−1)=y(1)=0, where p>q>−1 and λ>0 is a positive parameter. We discuss the existence of positive solutions and give a complete study.     

p-Catalan numbers and squarefree binomial coefficients

In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p^q divides F(p^q,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ₁(p^q) sufficiently large (τ₁(p^q) computable). Moreover, using the results of the first part, we find n<τ₁(p^q) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10.     

Class numbers of cyclotomic fields

In the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number h_p⁺ of the pth real cyclotomic field satisfies h_p⁺ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that $\text{h}{}_{\mathrm{n}}{}^{\mathrm{+}}\text{> n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; \epsilon </mtext>}}$ for infinitely many n. In the second part of the paper we show that if l ?= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then $\text{l}\text{2}$ (l, if l is odd) divides the class number h_n⁺ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides h_n⁺.     

Behavior at infinity of solutions of an equation of Osserman-Keller type

We study the behavior at infinity of solutions of equations of the form Δu=u^p, where p>1, in dimensions n?3. In particular we extend results proved by Loewner and Nirenberg in Contribution to Analysis, 1974, pp. 245-272 for the case p=(n+2)/(n−2), n?3, to values of p in the range p>n/(n−2), n?3.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Configurations in projective planes and quadrilateral-star Ramsey numbers

Some classes of configurations in projective planes with polarity are constructed. As the main result, lower bounds for the Ramsey numbers r(n)=r(C₄;K_1,n) are derived from these geometric structures, which improve some bounds due to Parsons about 30 years ago, and also yield a new class of optimal values: r(q²-2q+1)=q²-q+1 whenever q is a power of 2. Moreover, the constructions also imply a known result on C₄-K_1,n bipartite Ramsey numbers.     

On SA, CA, and GA numbers

Gronwall’s function G is defined for n>1 by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >10⁸⁵⁷⁶.     

Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation v_t=div(|∇v|^p−2∇v)−v^q in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t^−αw(|x|t^−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem

(0.1)     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

A Liouville type theorem for polyharmonic elliptic systems

In this paper, we consider the polyharmonic system m(−Δ)U=Vq,m(−Δ)V=Up in RN, for m>1, N>2m, with p?1, q?1, but not both equal to 1, where m(−Δ) is the polyharmonic operator. Set α=2m(q+1)/(pq−1), β=2m(p+1)/(pq−1), for α,β∈[(N−2)/2,N−2m), we prove the nonexistence of positive solutions.     

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.