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Register a new userA characterization of Sturmian sequences by indistinguishable asymptotic pairs
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We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full $mathbb{Z}$-shift are indistinguishable if the sets of occurrences of every pattern in each sequence coincide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillos theorem which characterizes Christoffel words. Furthermore, we provide a full characterization of indistinguishable asymptotic pairs on arbitrary alphabets using substitutions and biinfinite characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.
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Let $(X,d)$ be a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,ldots,a_{n})$ is called the isometric sequence of $(X,d)$. In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) $a_k=1$ for some $k$ with $2leq kleq n-2$; (ii) $a_k=2$ for some $k$ with $4leq kleq frac{1+sqrt{1+4n}}{2}$; (iii) $a_3=2$; (iv) $a_2=a_3=3$. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the $d$-dimensional Euclidean space with small $a_3$, which are analogue problems on a sets with few distinct triangles discussed by Epstein, Lott, Miller and Palsson.
For an integer $qge2$, a $q$-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of~$q$. In this article, $q$-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every $q$-recursive sequence is $q$-regular in the sense of Allouche and Shallit and that a $q$-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for $q$-recursive sequences are then obtained based on a general result on the asymptotic analysis of $q$-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Sterns diatomic sequence, the number of non-zero elements in some generalized Pascals triangle and the number of unbordered factors in the Thue--Morse sequence. For the first two sequences, our analysis even leads to precise formulae{} without error terms.
We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of words of $F$ of length $d$. An $F$-maximal bifix code of degree $d$ in a Sturmian set of words on an alphabet with $k$ letters has $(k-1)d+1$ elements. This generalizes the fact that a Sturmian set contains $(k-1)d+1$ words of length $d$. Moreover, given an infinite word $x$, if there is a finite maximal bifix code $X$ of degree $d$ such that $x$ has at most $d$ factors of length $d$ in $X$, then $x$ is ultimately periodic. Our main result states that any $F$-maximal bifix code of degree $d$ on the alphabet $A$ is the basis of a subgroup of index $d$ of the free group on~$A$.
It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.
Sources of quantum light, in particular correlated photon pairs that are indistinguishable in all degrees of freedom, are the fundamental resource that enables continuous-variable quantum computation and paradigms such as Gaussian boson sampling. Nanophotonic systems offer a scalable platform for implementing sources of indistinguishable correlated photon pairs. However, such sources have so far relied on the use of a single component, such as a single waveguide or a ring resonator, which offers limited ability to tune the spectral and temporal correlations between photons. Here, we demonstrate the use of a topological photonic system comprising a two-dimensional array of ring resonators to generate indistinguishable photon pairs with dynamically tunable spectral and temporal correlations. Specifically, we realize dual-pump spontaneous four-wave mixing in this array of silicon ring resonators that exhibits topological edge states. We show that the linear dispersion of the edge states over a broad bandwidth allows us to tune the correlations, and therefore, quantum interference between photons by simply tuning the two pump frequencies in the edge band. Furthermore, we demonstrate energy-time entanglement between generated photons. We also show that our topological source is inherently protected against fabrication disorders. Our results pave the way for scalable and tunable sources of squeezed light that are indispensable for quantum information processing using continuous variables.
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