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with initial data
$(x_1,x_2,x_3)=(0,x,1)$
, is eventually constant, and that its transit time and limit functions (of x) are unbounded and continuous, respectively. In this paper, we prove that for the slightly modified recursion
Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta (n^{1/4})$. We study the analogous problem in the $\mathbb {Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb {Z}_n$ is $\Theta (n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
Let
$h \geq 2$
be a positive integer. We introduce the concept of minimal restricted asymptotic bases and obtain some examples of minimal restricted asymptotic bases of order h.
Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem’s strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.
Let
$\mathcal {A}$
be the set of all integers of the form
$\gcd (n, F_n)$
, where n is a positive integer and
$F_n$
denotes the nth Fibonacci number. Leonetti and Sanna proved that
$\mathcal {A}$
has natural density equal to zero, and asked for a more precise upper bound. We prove that
for all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.
A Sidon set is a subset of an Abelian group with the property that the sums of two distinct elements are distinct. We relate the Sidon sets constructed by Bose to affine subspaces of
$ \mathbb {F} _ {q ^ 2} $
of dimension one. We define Sidon arrays which are combinatorial objects giving a partition of the group
$\mathbb {Z}_{q ^ 2} $
as a union of Sidon sets. We also use linear recurring sequences to quickly obtain Bose-type Sidon sets without the need to use the discrete logarithm.
We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including
Recently, Lin and Liu [‘Congruences for the truncated Appell series
$F_3$
and
$F_4$
’, Integral Transforms Spec. Funct.31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series
$F_3$
. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.
When a page, represented by the interval
$[0,1]$
, is folded right over left
$n $
times, the right-hand fold contains a sequence of points. We specify these points using two different representation techniques, both involving binary signed-digit representations.
Let n and k be positive integers with
$n\ge k+1$
and let
$\{a_i\}_{i=1}^n$
be a strictly increasing sequence of positive integers. Let
$S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$
. In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull.20 (1978), 117–118] confirmed a conjecture of Erdős by showing that
$S_{n,1}\le 1-{1}/{2^{n-1}}$
. Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar.160 (2020), 360–375] improved Borwein’s upper bound to
$S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$
and derived optimal upper bounds for
$S_{n,2}$
and
$S_{n,3}$
. In this paper, we present a sharp upper bound for
$S_{n,4}$
and characterise the sequences
$\{a_i\}_{i=1}^n$
for which the upper bound is attained.
We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci.2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math.72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$. Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$. In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358].
Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the
$\mathbb{C}$
-algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.
By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart.3(2) (1965), 81–89].
We consider sums involving the divisor function over nonhomogeneous (
$\beta \neq 0$
) Beatty sequences
$ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $
and show that
where N is a sufficiently large integer,
$\alpha $
is of finite type
$\tau $
and
$\beta \neq 0$
. Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all
$\alpha $
.
Let
$\mathbb {N}$
be the set of all nonnegative integers. For
$S\subseteq \mathbb {N}$
and
$n\in \mathbb {N}$
, let
$R_S(n)$
denote the number of solutions of the equation
$n=s_1+s_2$
,
$s_1,s_2\in S$
and
$s_1<s_2$
. Let A be the set of all nonnegative integers which contain an even number of digits
$1$
in their binary representations and
$B=\mathbb {N}\setminus A$
. Put
$A_l=A\cap [0,2^l-1]$
and
$B_l=B\cap [0,2^l-1]$
. We prove that if
$C \cup D=[0, m]\setminus \{r\}$
with
$0<r<m$
,
$C \cap D=\emptyset $
and
$0 \in C$
, then
$R_{C}(n)=R_{D}(n)$
for any nonnegative integer n if and only if there exists an integer
$l \geq 1$
such that
$m=2^{l}$
,
$r=2^{l-1}$
,
$C=A_{l-1} \cup (2^{l-1}+1+B_{l-1})$
and
$D=B_{l-1} \cup (2^{l-1}+1+A_{l-1})$
. Kiss and Sándor [‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math.340 (2017), 1154–1161] proved an analogous result when
$C\cup D=[0,m]$
,
$0\in C$
and
$C\cap D=\{r\}$
.
For a set A of positive integers and any positive integer n, let
$R_{1}(A, n)$
,
$R_{2}(A,n)$
and
$R_{3}(A,n)$
denote the number of solutions of
$a+a^{\prime }=n$
with
$a, a^{\prime }\in A$
and the additional restriction that
$a<a^{\prime }$
for
$R_{2}$
and
$a\leq a^{\prime }$
for
$R_{3}$
. We consider Problem 6 of Erdős et al. [‘On additive properties of general sequences’, Discrete Math.136 (1994), 75–99] about locally small and locally large values of
$R_{1}, R_{2}$
and
$R_{3}$
.
Let
$k\geq 2$
be an integer. We prove that the 2-automatic sequence of odious numbers
$\mathcal {O}$
is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation
$f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$
for arbitrary
$x_1,\ldots ,x_k\in \mathcal {O}$
, then f is the identity function
$f(n)=n$
for all
$n\in \mathbb {N}$
.