On the Entropy of Multidimensional Multiplicative Integer Subshifts

Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$

“…Then we establish LDP of multiple average (13) with the rate function I r (y) = ηy − F r (η), if F r (η) = y, by the differentiability of F r (β) and Theorem 3.1. In section 4, the LDP of the weighted multiple average ( 16) is also established for r = 1 2 . Note that, when r = 1 2 , the free energy funciotn is difficult to compute since the matrices are not commute in general.…”

“…Later, the multifractal analysis and the dimension theory of the multiple ergodic averages AnF(x) N in N (or Z) are also interesting research subjects and have been studied in depth recently (cf. [1,2,5,12,13,17,18,19,20]). We also refer the reader to [10] for the survey of this subject and find the complete bibliography therein.…”

“…It is obvious that X p1,p 2 ,...,p k−1 is a N d version of X p1,p2,...,p k−1 . Recently, Ban, Hu and Lai [1] established the Minkowski dimension of (4). Related works on the dimension theory of the multidimensional multiple sum can also be found in [5].…”

“…The difficulty is that it is not easy to decompose the multidimensional lattices into independent sublattices according to the given 'multiple constraints', e.g., the p i s in (4), and calculate its density among the entire lattice. Fortunately, the technique developed in [1] is useful and leads us to investigate the LDP for the multidimensional multiple averages launched by Carinci et al in 2012 [6], and multidimensional weighted multiple sum mentioned in [11]. Both topics are described in the following two paragraphs.…”

“…3. When r = 1/2, we have h = 1 2 log(r/1 − r) = 0. Then for all β ∈ R, the vector v is not parallel to the vector w + and so is e + , that gives…”