A short proof of Erd\H{o}s's $B+C$ conjecture
We give a short proof of the fact that every set of natural numbers with positive upper Banach density contains the sum of two infinite sets. The approach simplifies earlier existing proofs.
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arXiv subjects
Donald Robertson
Publications and source records attributed to Donald Robertson.
Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits
For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.
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A structure theorem for polynomial return-time sets in minimal systems
We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent.
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The Density Finite Sums Theorem
For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$ for all $m\in\{1,\ldots,k\}$. This can be viewed as a density analog of Hindman's finite sums theorem. Our proof reveals the natural relationships among infinite sumsets, the dynamics underpinning arithmetic progressions, and homogeneous spaces of nilpotent Lie groups.
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A rank one mild mixing system without minimal self joinings
We show that there is a rank 1 transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.
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Problems on infinite sumset configurations in the integers and beyond
In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density.
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A proof of Erd\H{o}s's $B+B+t$ conjecture
We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.
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Infinite Sumsets in Sets with Positive Density
Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ 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 $k=2$.
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Determinantal formulas with major indices
We give a simple proof of a major index determinant formula in the symmetric group discovered by Krattenthaler and first proved by Thibon using noncommutative symmetric functions. We do so by proving a factorization of an element in the group ring of the symmetric group. By applying similar methods to the groups of signed permutations and colored permutations, we prove determinant formulas in these groups as conjectured by Krattenthaler.
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Unique equilibrium states for some intermediate beta transformations
We prove uniqueness of equilibrium states for subshifts corresponding to intermediate beta transformations with $\beta > 2$ having the property that the orbit of 0 is bounded away from 1.
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Uniform distribution of saddle connection lengths
For almost every flat surface the sequence of saddle connection lengths listed in increasing order is uniformly distributed mod one.
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A proof of a sumset conjecture of Erd\H{o}s
In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.
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Ergodicity of skew products over linearly recurrent IETs
We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure.
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Disjointness for measurably distal group actions and applications
We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence.
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Mild mixing of certain interval exchange transformations
We prove that irreducible, linearly recurrent, type W interval exchange transformations are always mild mixing. For every irreducible permutation the set of linearly recurrent interval exchange transformations has full Hausdorff dimension.
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Triangles in cartesian squares of quasirandom groups
We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.
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Polynomial multiple recurrence over rings of integers
We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^m$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne on polynomials over the integers.
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Polynomial recurrence with large intersection over countable fields
We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong recurrence times.
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