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We discuss properties of integers in base 3/2. We also introduce many new sequences related to base 3/2. Some sequences discuss patterns related to integers in base 3/2. Other sequence are analogues of famous base-10 sequences: we discuss powers of 3 and 2, Look-and-say, and sorted and reverse sorted Fibonaccis. The eventual behavior of sorted and reverse sorted Fibs leads to special Pinocchio and Oihcconip sequences respectively.
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We delve into the connection between base $frac{3}{2}$ and the greedy partition of non-negative integers into 3-free sequences. Specifically, we find a fractal structure on strings written with digits 0, 1, and 2. We use this structure to prove that the even non-negative integers written in base $frac{3}{2}$ and then interpreted in base 3 form the Stanley cross-sequence, where the Stanley cross-sequence comprises the first terms of the infinitely many sequences that are formed by the greedy partition of non-negative integers into 3-free sequences.
We give a characterization of all matrices $A,B,C in mathbb{F}_{2}^{m times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,Cinmathbb{F}_{2}^{mathbb{N}timesmathbb{N}}$ which generate a $(0,2)$-sequence in base $2$.
We discuss two different systems of number representations that both can be called base 3/2. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.
Let $mathcal{I} subset mathbb{N}$ be an infinite subset, and let ${a_i}_{i in mathcal{I}}$ be a sequence of nonzero real numbers indexed by $mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| leq C_1 cdot i^m$ for all $i in mathcal{I}$. Furthermore, let $c_i in [-1,1]$ be defined by $c_i = frac{a_i}{C_1 cdot i^m}$ for each $i in mathcal{I}$, and suppose the $c_i$s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $mu$. In this paper, we show that if $mathcal{I} subset mathbb{N}$ is not too sparse, then the sequence ${a_i}_{i in mathcal{I}}$ fails to obey Benfords Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $mu([0,t])$ is a strictly convex function of $t in (0,1)$. Nonetheless, we also provide conditions on the density of $mathcal{I} subset mathbb{N}$ under which the sequence ${a_i}_{i in mathcal{I}}$ satisfies Benfords Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benfords Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.
In 2009, Grant, Shallit, and Stoll constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.
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