Some ergodic theorems involving Omega function and their applications

On the interplay between additive and multiplicative largeness and its combinatorial applications

The Furstenberg–Sárközy theorem and asymptotic total ergodicity phenomena in modular rings

Chapter 12 Combinatorial and diophantine applications of ergodic theory

In this note we give a short, direct proof of the combinatorial Nullstellensatz.

The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{v}$</tex></formula> can be divided into classical cyclotomy or generalized cyclotomy, depending on <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$v$</tex> </formula> prime or composite. In this paper, we introduce a generalized cyclotomy of order <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex></formula> over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots, p_{n}^{e_{n}}}$</tex></formula> , which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p_{1},p_{2},\ldots,p_{n}$</tex></formula> are pairwise distinct odd primes satisfying <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d\vert (p_{i}-1)$</tex></formula> for all <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1\leq i\leq n$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$e_{1},e_{2},\ldots,e_{n}$</tex></formula> are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{p_{1}p_{2}}$</tex> </formula> . Besides, we also construct an infinite series of near-optimal codebooks over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\BBZ}_{p_{1}p_{2}}$</tex></formula> , as well as some infinite series of asymptotically optimal difference systems of sets over <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},\ldots,p_{n}^{e_{n}}}$</tex> </formula> .

Motivated by questions asked by Erdős, we prove that any set A ⊂ N A\subset \mathbb {N} with positive upper density contains, for any k ∈ N k\in \mathbb {N} , a sumset B 1 + ⋯ + B k B_1+\cdots +B_k , where B 1 B_1 , …, B k ⊂ N B_k\subset \mathbb {N} are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of k = 2 k=2 .

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measure-theoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from ℝd to [0, 1], which we view as a compact subspace of L ∞(ℝd) in the weak-* topology. The pointwise ergodic theorem for ℝd-actions implies that with respect to any translation-invariant measure on this space, almost every function is asymptotically close to a constant function at large scales. This observation leads to a general sufficient condition for a configuration to occur in every set of positive upper Banach density at all sufficiently large scales. To illustrate the use of this criterion we apply it to prove a new result concerning three-point configurations in measurable subsets of the plane which form the vertices of a triangle with specified area and side length, yielding a new proof of a result related to work of R. Graham.

The paper is concerned with sums of the type \[S_{n,j} = \sum {x_1^{a_1 } x_2^{a_2 } \cdots x_n^{a_n } } \quad (n > 1),\] where the summation is over either \[( * )\qquad ja_i \leqq a_1 + a_2 + \cdots + a_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n)\] or \[( * * )\qquad a_1 + a_2 + \cdots + a_n = ja_i + (n - j)b_i \quad (0 \leqq j \leqq n;1 \leqq i \leqq n);\] the $a_i $ and $b_i $ are nonnegative integers. It is proved, for example, that for the first type with $j = 2$, the sum is a rational function with denominator equal to $\prod _{1 \leqq i < k \leqq n} (1 - x_i x_k )$. Several combinatorial applications are obtained by specializing the $x_i $. For example it is proved that the number of nonnegative solutions of the system \[a_1 + a_2 + \cdots + a_n = N,\quad (n - 1)a_i \leqq N,\quad (1 \leqq i \leqq n)\] is equal to the binomial coefficient \[\left( {\begin{array}{*{20}c} {k + n - s - 1} \\ {n - 1} \\ \end{array} } \right)\quad (N = k(n - 1) + s,0 \leqq s < n - 1).\] The final section of the paper is concerned with multiple Dirichlet series \[{\bf \Phi} _{n,j} = \sum {m_1^{ - s_1 } m_2^{ - s_2 } \cdots m_n^{ - s_n } } ,\] where the smmation is over all positive integers $m_i $ such that \[m_i^j \mid m_1 m_2 \cdots m_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n).\] The ${\bf \Phi} _{n,j} $ are expressed as products involving series satisfying (*); in particular \[{\bf \Phi} _{n,n - 1} = \frac{{\zeta (\alpha )\zeta (\sigma - s_1 ) \cdots \zeta (\alpha - s_n )}}{{\zeta ((n - 1)\sigma )}},\] where $\sigma = s_1 + \cdots + s_n $ and $\zeta (s)$ is the Riemann zeta-function.

Erdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman (Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems, 25 (2005) no. 1, 201-213) and Frantzikinakis (Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d'Analyse Mathematique, 112 (2010), 79-135 and Equidistribution of sparse sequences on nilmanifolds. Journal d'Analyse Mathematique, 109 (2009), 353-395) we show that the ergodic averages with iterates given by the integer part of real-valued strongly independent polynomials, converge in the mean to the right-expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications while combining these results with methods from The polynomial multidimensional Szemeredi theorem along shifted primes. Israel J. Math., 194 (2013), no. 1, 331-348 and Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems, 1-20. doi:10.1017/etds.2016.40 we get the respective right limits and combinatorial results for multiple averages for a single sequence as well as for several sequences along prime numbers.

This paper investigates the structure and properties of block-symmetric and block-super\-symmetric polynomials in Banach spaces. The study extends classical symmetric polynomial results to infinite-dimensional settings, particularly in sequence spaces such as $\ell_p(\mathbb{C}^s),$ $1\leq p&lt;\infty$ and spaces of two-sided absolutely summing series of vectors in $\mathbb{C}^s$ for some positive integer $s&gt;1.$ In this paper, we derive analogs of the Waring-Girard formulas for block-symmetric and block-supersymmetric polynomials and explore their combinatorial applications.

Eulerian numbers, Newcomb's problem and representations of symmetric groups

Shortly after Szemeredi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of ergodic Ramsey Theory, in which problems motivated by additive combinatorics are proven using ergodic theory. Ergodic Ramsey Theory has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure preserving systems. We outline the ergodic theory background needed to understand these results, with an emphasis on recent developments in ergodic theory and the relation to recent developments in additive combinatorics. These notes are based on four lectures given during the School on Additive Combinatorics at the Centre de Recherches Mathematiques, Montreal in April, 2006. The talks were aimed at an audience without background in ergodic theory. No attempt is made to include complete proofs of all statements and often the reader is referred to the original sources. Many of the proofs included are classic, included as an indication of which ingredients play a role in the developments of the past ten years.

On uniformly distributed sequences of integers and Poincaré recurrence

On uniformly distributed sequences of integers and Poincaré recurrence II