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Register a new userOn Furstenberg systems of aperiodic multiplicative functions of Matomaki, Radziwill and Tao
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It is shown that in a class of counterexamples to Elliotts conjecture by Matomaki, Radziwill and Tao, the Chowla conjecture holds along a subsequence.
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We show that Sarnaks conjecture on Mobius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $PinR[x]$ with irrational leading coefficient and for each multiplicative function $bnu:NtoC$, $|bnu|leq1$, we have[ frac{1}{M} sum_{Mle mtextless{}2M} frac{1}{H} left| sum_{mle n textless{} m+H} e^{2pi iP(n)}bnu(n) right|longrightarrow 0 ] as $Mtoinfty$, $Htoinfty$, $H/Mto 0$.
We study sets of recurrence, in both measurable and topological settings, for actions of $(mathbb{N},times)$ and $(mathbb{Q}^{>0},times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form ${(an+b)^{ell}/(cn+d)^{ell}: n in mathbb{N}}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $log r / log s$ is irrational and $X$ and $Y$ are $times r$- and $times s$-invariant subsets of $[0,1]$, respectively, then $dim_text{H} (X+Y) = min ( 1, dim_text{H} X + dim_text{H} Y)$. Our main result yields information on the size of the sumset $lambda X + eta Y$ uniformly across a compact set of parameters at fixed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.
We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.
This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $beta$-expansions. More precisely, let $([0,1),T_{beta})$ be the $beta$-dynamical system for a general $beta>1$ and $psi:[0,1]mapstomathbb{R}$ be a continuous function. Denote by $textsf{A}(psi,x)$ all the accumulation points of $Big{frac{1}{n}sum_{j=0}^{n-1}psi(T^jx): nge 1Big}$. The Hausdorff dimensions of the sets $$Big{x:textsf{A}(psi,x)supset[a,b]Big}, Big{x:textsf{A}(psi,x)=[a,b]Big}, Big{x:textsf{A}(psi,x)subset[a,b]Big}$$ i.e., the points for which the Birkhoff averages of $psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $psi$.
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