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Register a new userCriteria for regularity of Mahler power series and Beckers conjecture
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Authors
Tomasz Kisielewski
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Allouche and Shallit introduced the notion of a regular power series as a generalization of automatic sequences. Becker showed that all regular power series satisfy Mahler equations and conjectured equivalent conditions for the converse to be true. We prove a stronger form of Beckers conjecture for a subclass of Mahler power series.
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In 1998, Allouche, Peyri`{e}re, Wen and Wen showed that the Hankel determinant $H_n$ of the Thue-Morse sequence over ${-1,1}$ satisfies $H_n/2^{n-1}equiv 1~(mathrm{mod}~2)$ for all $ngeq 1$. Inspired by this result, Fu and Han introduced emph{apwenian} sequences over ${-1,1}$, namely, $pm 1$ sequences whose Hankel determinants satisfy $H_n/2^{n-1}equiv 1~(mathrm{mod}~2)$ for all $ngeq 1$, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let $f(z)$ be the generating functions of such apwenian sequences. We show that for all integer $bge 2$ with $f(1/b) eq 0$, the real number $f(1/b)$ is transcendental and its irrationality exponent is equal to $2$. Besides, we also derive a criterion for zero-one apwenian sequences whose Hankel determinants satisfy $H_nequiv 1~(mathrm{mod}~2)$ for all $ngeq 1$. We find that the only zero-one apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also given. Furthermore, we prove that all Sturmian sequences over ${-1,1}$ or ${0,1}$ are not apwenian. And we conjecture that fixed points of substitution of non-constant length over ${-1,1}$ or ${0,1}$ can not be apwenian.
It is a classical result of Mahler that for any rational number $alpha$ > 1 which is not an integer and any real 0 < c < 1, the set of positive integers n such that $alpha$ n < c n is necessarily finite. Here for any real x, x denotes the distance from its nearest integer. The problem of classifying all real algebraic numbers greater than one exhibiting the above phenomenon was suggested by Mahler. This was solved by a beautiful work of Corvaja and Zannier. On the other hand, for non-zero real numbers $lambda$ and $alpha$ with $alpha$ > 1, Hardy about a century ago asked In what circumstances can it be true that $lambda$$alpha$ n $rightarrow$ 0 as n $rightarrow$ $infty$? This question is still open in general. In this note, we study its analogue in the context of the problem of Mahler. We first compare and contrast with what is known visa -vis the original question of Hardy. We then suggest a number of questions that arise as natural consequences of our investigation. Of these questions, we answer one and offer some insight into others.
In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $eta(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $eta(G)$. This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.
We identify a class of semi-modular forms invariant on special subgroups of $GL_2(mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.
The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by ${s_{m,n}}_{ngeq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)sec (btx)(pm cos ((b-p)tx)pm sin (ptx))$, where p is an integer satisfying $0leq pleq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.
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