Giuseppe Melfi
A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.
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Aperiodicity and subword complexity in the binary expansion of powers of three
研究三的幂次二进制展开的精细结构，证明对于固定周期\(p\)，\(3^{m}\)二进制展开打破\(p\) - 周期性的位置数增长阶为\(\log m / \log\log m\)，且由\(3^{m}\)低阶数字形成的二进制字具有完整低阶子词复杂性。
Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Waring problem in numbers of class $\mathbb{N}_0$ is solved.
Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Goldbach's problem in numbers of class $\mathbb{N}_0$ is solved.
Kaplansky [2003] proved a theorem on the simultaneous representation of a prime $p$ by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink's list. We also show how such theorems are related to results of Muskat on binary quadratic forms
We study the largest values of the $r$th row of Stern's diatomic array. In particular, we prove some conjectures of Lansing. Our main tool is the connection between the Stern sequence, alternating binary expansions and continuants. This allows us to reduce the problem of ordering the elements of the Stern sequence to the problem of ordering continuants. We describe an operation that increases the value of a continuant, allowing us to reduce the problem of largest continuants to ordering continuants of very special shape. Finally, we order these special continuants using some identities and ine…
Block occurrences in the binary expansion
本文研究二进制展开中块 $\mathtt{11}$ 出现次数的差值，证明其概率分布均匀接近高斯，误差随 $t$ 中 1 的块数增加而趋于 0。
The Binary Two-Up Sequence is the lexicographically earliest sequence of distinct nonnegative integers with the property that the binary expansion of the n-th term has no 1-bits in common with any of the previous floor(n/2) terms. We show that the sequence can be decomposed into ``atoms'', which are sequences of 4, 6, or 8 numbers whose binary expansions match certain patterns, and that the sequence is the limiting form of a certain ``word'' involving the atoms. This leads to a fairly explicit formula for the terms, and in particular establishes the conjecture that every nonzero term is the su…
The Stern-Brocot tree and Minkowki's question mark function $?(x)$ (or Conway's box function) are related to the continued fraction expansion of numbers from Q with unary encoding of the partial denominators. We first define binary encodings $C_I, C_{II}$ of the natural numbers, adapted to the Gauß-Kuz'min measure for the distribution of partial denominators. We then define the V$_1$ tree as analogue to the Stern-Brocot tree, using the binary encondings $C_I, C_{II}$. We shall see that all numbers with denominator $q$ are present in the first $3.44\log_2(q)$ levels, instead of $1/q$ appearing …
Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
本文建立了二元四次型与三元二次型对协变间的对应，并给出了任意特征零域上二次型对的标准对角化方法及有理数域上可对角化的精确判据。
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way from what we call "iteration sequences". We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of nonoccurrence of certain finite blocks, a…
On the behavior of binary block-counting functions under addition
本文将二进制数字和差值分布的研究推广至二进制分块计数函数族 $N^w$，证明了当 $t$ 的二进制展开包含足够多 $\mathtt{1}$s 块时，差值 $N^w(n+t) - N^w(n)$ 的分布近似于高斯分布。
The binary sum-of-digits function $s$ counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer $t$, T.~W.~Cusick defined the asymptotic density $c_t$ of integers $n\geq 0$ such that \[s(n+t)\geq s(n).\] In 2011, he conjectured that $c_t>1/2$ for all $t$ -- the binary sum of digits should, more often than not, weakly increase when a constant is added. In this paper, we prove that there exists an explicit constant $M_0$ such that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M_0$ maximal blocks of contiguous ones, leaving open o…
We show that the number of $1$'s in the first $N$ digits of the binary expansion of $\sqrt{2}$ is at least $\sqrt{2N}(1+o(1))$ and show that this bound can be improved to around $2\sqrt{N}/\sqrt{2\sqrt{2}-1}$ infinitely often.
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < ε$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.
On a family of continued fractions in $Q((T^1))$ associated to infinite binary words derived from the Thue-Morse sequence
本文针对每个整数n>1，在$Q((T^{-1}))$中构造元素，其连分数展开具有特定模式，且与由Thue-Morse序列类似定义的无限二进制词W(n)相关。
A palindrome in base $g$ is an integer $N$ that remains the same when its digit expansion in base $g$ is reversed. Let $g$ and $h$ be given distinct integers $>1$. In this paper we discuss how many integers are palindromes in base $g$ and simultaneously palindromes in base $h$.
A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and…
Hausdorff dimension of double base expansions and binary shifts with a hole
本文研究了双基展开和二进制移位中存在孔洞的动力系统，推导了豪斯多夫维数和熵的显式公式，并证明了其连续性。
We present a new result on counting primes $p<N=2^n$ for which $r$ (arbitrarily placed) digits in the binary expansion of $p$ are specified. Compared with earlier work of Harman and Katai, the restriction on $r$ is relaxed to $r< c\Big(\frac n{\log n}\Big)^{4/7}$. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of $L$-functions with `powerful' conductor.
The sequence A268289 from the On-Line Encyclopedia of Integer Sequences, namely the cumulated differences between the number of digits 1 and the number of digits 0 in the binary expansion of consecutive integers, is studied here. This sequence happens to match a sequence of cardinalities of some specific sets. Furthermore, it can also be expressed by using the Takagi function. The three different definitions have their own properties and combining them together lead to some new identities.
Accelerating true orbit pseudorandom number generation using Newton's method
本研究针对无理代数数二进制展开的伪随机序列生成问题，提出基于牛顿法的高效计算方法，证明其时间复杂度与N位整数乘法相当，加速了真轨道伪随机数生成且序列通过TestU01的RabbitFile全部统计测试。
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