Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

Clarkson-Erdős theorem for many variables

In the present paper, the Clarkson-Erdős, which complements the well-known Müntz theorem, is extended to the case of many variables. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 594–598, April, 1999.     

Hua’s theorem with nine almost equal prime variables

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as

Hardy’s theorem and rotations of several complex variables

In this paper, Hardy’s theorem and rotations characterized by complex Gaussians in the complex plane due to Hogan and Lakey are extended to complex spaces of several variables. We point out that conditions under which a function on the n-dimensional real Euclidean space has an analytic extension to the complex space. Moreover, we prove that the function is a rotation of a multiple of real Gaussians through some angle if the extension satisfies certain assumptions.     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

Engel’s theorem for malcev superalgebras

A version of Engel’s theorem for Malcev superalgebras is proved in the spirit of theJacobson-Engel theorem for Lie algebras. Some consequences for the structure of Malcev superalgebras with trivial Lie nucleus are derived.     

Menger’s theorem for infinite graphs

We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set of disjoint A–B paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in . This settles an old conjecture of Erdős.     

Moufang’s theorem for non-Moufang loops

We introduce a class of non-Moufang loops satisfying Moufang’s theorem.     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Ando’s theorem for nonnegative forms

In this paper, we present a generalization of Ando??s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et?al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1?C12, 1999). That is, [A]B??? [B]A or [B]A??? [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et?al. in J. Funct. Anal. 257(12), 3858?C3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.     

Alperin’s fusion theorem for localities

In these notes we give a version of the Alperin–Goldschmidt fusion theorem for localities.     

Permutation-invariance in Komlós type theorem for non-negative random variables

We provide a permutation-invariant version of Komlós’ type convergence for non-negative random variables.     

Chevalley��s theorem with restricted variables

First, a generalization of Chevalley’s classical theorem from 1936 on polynomial equations f(x ₁,...,x _N ) = 0 over a finite field K is given, where the variables x _i are restricted to arbitrary subsets A _i ⊆ K. The proof uses Alon’s Nullstellensatz. Next, a theorem on integer polynomial congruences f(x ₁,...,x _N ) ≡ 0 (mod p ^v ) with restricted variables is proved, which generalizes a more recent result of Schanuel. Finally, an extension of Olson’s theorem on zero-sum sequences in finite Abelian p-groups is derived as a corollary.     

An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ? ^m that are continuous on the whole space ? ^m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f _k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f _k are complex-valued.     

Beurling’s analyticity theorem for quantum differences

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ _n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.     

Lazard’s theorem for S-pure flat modules

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.     

Naishul’s theorem for fibered holomorphic maps

We show that the fibered rotation number associated to an indifferent invariant curve for a fibered holomorphic map is a topological invariant.     

Hechler’s theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechlers classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by Bartoszyski and Kada.Research supported by NSERC. The first author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when the present paper was completed.The second author was supported by Grant-in-Aid for Young Scientists (B) 14740058, MEXT.Mathematics Subject Classification (2000): 03E35, 03E17Revised version: 16 February 2004