A szemerédi type theorem for sets of positive density inR k

Letk≧2 andA a subset ofR ^k of positive upper density. LetV be the set of vertices of a (non-degenerate) (k−1)-dimensional simplex. It is shown that there existsl=l(A, V) such thatA contains an isometric image ofl′. V wheneverl′>l. The casek=2 yields a new proof of a result of Katznelson and Weiss [4]. Using related ideas, a proof is given of Roth’s theorem on the existence of arithmetic progressions of length 3 in sets of positive density.     

Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.     

Problems of almost everywhere convergence related to harmonic analysis and number theory

The object of this paper is to discuss certain methods for studying almost everywhere convergence problems. We consider the generalization of the Riesz-Raikov theorem where the dilation numberϑ>1 is not necessarily an integer. It is known (see [B2]) that the averages (1/N)Σ ₁ ^N f(ϑ ⁿ x) converge a.e. to ∝ ₀ ¹ fdx wheneverϑ is algebraic andf a 1-periodic function onR satisfying ∝ ₀ ¹ |f(x)|² dx<∞. Here the particular case of rational dilation is treated. The reader is referred to [B2] for the general (algebraic) case. The following definitive relation between a.e. convergence and algebraic numbers is proved. Let {μ _j} be the sequence of measures converging weak* to the natural measureμ on the Cantor set of dissection ratioϑ. Thenf*μ _j→f*μ a.e. for allL ^∞ (T) functions iffϑ is algebraic. This fact depends on [B3] and a variant of Rota’s theorem [Ro] on a.e. convergence of certain compositions of operators. Further applications of this result in ergodic theory are presented in the last section of the paper. In section 4, a.e. convergence of Riemann sums of periodicL ²-functions is investigated. It is shown that almost surelyR _n f has a logarithmic density, where . This result complements the work of R. Salem on the subject.     

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

Communications in Mathematical Physics - We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo (Forum Math, 2014) about regions $${\mathcal{R}_g \subset \mathbb{C}}$$ for...     

On the spectral gap for finitely-generated subgroups of SU(2)

We prove the spectral gap property for free subgroups of SU(2) generated by elements satisfying a noncommutative diophantine property, in particular for free subgroups generated by elements with algebraic entries.     

On the correlation of the Moebius functionwith rank-one systems

We explore the “Moebius disjointness property” in the special context of rank-one transformations and verify this phenomenon for many of the “classical” models.     

On the Hardy-Littlewood maximal function for the cube

In this note, we are interested in local-global principles for multinorm equations \(\prod\nolimits_{i = 1}^n {{N_{{L_i}/k}}({z_i}} ) = a\) where k is a global field, L _i /k are finite separable field extensions and a ∈ k*. In particular, we prove a result relating the Hasse principle and weak approximation for this equation to the Hasse principle and weak approximation for some classical norm equation N _F/k (w) = a where \(F: = \bigcap\nolimits_{i = 1}^n {{L_i}} \) . It provides a proof of a “weak approximation” analogue of a recent conjecture by Pollio and Rapinchuk about the multinorm principle. We also provide a counterexample to the original conjecture concerning the Hasse principle.