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Idris Assani

Publications and source records attributed to Idris Assani.

23 records · Page 2Linked to original sources
arXiv2005

Averages along cubes for not necessarily commuting measure preserving transformations

We study the pointwise convergence of some weighted averages linked to averages along cubes. We show that if $(X,\mathcal{B},μ, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$ $1\leq i\leq 6$ are bounded functions then the averages $$\frac{1}{N^3}\sum_{n, m, p=1}^N f_1(T_1^nx) f_2(T_2^mx) f_3(T_3^px) f_4(T_4^{n+m}x) f_5(T_5^{n+p}x) f_6(T_6^{m+p}x)$$ converge almost everywhere.

math.DS↗
arXiv2003

Pointwise convergence of averages along cubes

Let $(X,\mathcal{B},μ, T)$ be a measure preserving system. We prove the pointwise convergence of the averages $$\frac{1}{N^2}\sum_{n,m= 0}^{N-1} f_1(T^nx)f_2(T^mx)f_3(T^{n+m}x)$$ and of similar averages with seven bounded functions.

math.DS↗
arXiv2003

Pointwise convergence along cubes for measure preserving systems

Let $(X, \mathcal{B}, μ)$ be a probability measure space and $T_1$, $T_2$, $T_3$ three not necessarily commuting measure preserving transformations on $(X, \mathcal{B}, μ)$. We prove that for all bounded functions $f_1$, $f_2$, $f_3$ the averages $$\frac{1}{N^2}\sum_{n, m =1}^N f_1(T_1^nx)f_2(T_2^mx)f_3(T_3^{n+m}x)$$ converges a.e. Generalizations to averages of $2^k -1$ functions are also given for not necessarily commuting weakly mixing systems.

math.DS↗
arXiv2003

An $L^1$ counting problem in ergodic theory

We solve the following counting problem for measure preserving transformations. For $f\in L_+^1(μ)$, is it true that $\ds \sup_n\frac{\bN_n(f)(x)}{n} <\infty,$ where $$\ds\bN_n(f)(x)= # {k: \frac{f(T^k x)}{k}>\frac 1 n}?$$ One of the consequences is the nonvalidity of J. Bourgain's Return Time Theorem for pairs of $(L^1, L^1)$ functions.

math.DS↗