Multiple recurrence and large intersections for abelian group actions

The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $varphi, psi : G to G$ are homomorphisms such that $varphi(G)$, $psi(G)$, and $(psi - varphi)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{varphi(g)}^{-1}A cap T_{psi(g)}^{-1}A) > mu(A)^3 - varepsilon}$ is syndetic. (2) If $G$ is a countable abelian group and $r,s in mathbb{Z}$ are integers such that $rG$, $sG$, and $(r pm s)G$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{rg}^{-1}A cap T_{sg}^{-1}A cap T_{(r+s)g}^{-1}A) > mu(A)^4 - varepsilon}$ is syndetic. In particular, these extend and generalize results of Bergelson, Host, and Kra concerning $mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $mathbb{F}_p^{infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. We also discuss numerous examples shedding light on the necessity of the various hypotheses above. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper.

The Furstenberg-Sarkozy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

The Furstenberg-Sarkozy theorem asserts that the difference set $E-E$ of a subset $E subset mathbb{N}$ with positive upper density intersects the image set of any polynomial $P in mathbb{Z}[n]$ for which $P(0)=0$. Furstenbergs approach relies on a correspondence principle and a polynomial version of the Poincare recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,mathcal{B},mu,T)$ and set $A in mathcal{B}$ with $mu(A) > 0$, one has $c(A):= lim_{N to infty} frac{1}{N} sum_{n=1}^N mu(A cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $mu(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $mathbb{Z}/Nmathbb{Z}$. We show that a sequence of modular rings $mathbb{Z}/N_mmathbb{Z}$, $m in mathbb{N},$ is asymptotically totally ergodic if and only if $mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $delta in (0,1]$ and a (not necessarily intersective) polynomial $Q in mathbb{Q}[n]$ such that $Q(mathbb{Z}) subseteq mathbb{Z}$, and write $S = { Q(n) : n in mathbb{Z}/Nmathbb{Z}}$. For any integer $N > 1$ with $mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $mathbb{Z}/Nmathbb{Z}$ such that $|A||B| geq delta N^2$, then $mathbb{Z}/Nmathbb{Z} = A + B + S$.

Sums of Powers in Large Finite Fields: A Mix of Methods

Can any element in a sufficiently large finite field be represented as a sum of two $d$th powers in the field? In this article, we recount some of the history of this problem, touching on cyclotomy, Fermats last theorem, and diagonal equations. Then, we offer two proofs, one new and elementary, and the other more classical, based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. In context and juxtaposition, each will have its merits.

The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) in mathbb{Z}[x]$; for each integer $alpha$ it is interesting to consider the number of iterates $n_{alpha}$, if possible, needed to satisfy $f^{n_{alpha}}(alpha) = alpha$. The sets ${alpha, f(alpha), ldots, f^{n_{alpha} - 1}(alpha), alpha}$ generated by the iterates of $f$ are called cycles. For $mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $mathbb{Z}$ by adjoining reciprocals of primes. Let $mathbb{Z}[1/p_1, ldots, 1/p_n]$ denote $mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, ldots, p_n$ and all their products and powers with each other and the elements of $mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $mathbb{Z}left[1/p_1, ldots, 1/p_nright][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.