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We examine a recursive sequence in which
$s_n$
is a literal description of what the binary expansion of the previous term
$s_{n-1}$
is not. By adapting a technique of Conway, we determine the limiting behaviour of
$\{s_n\}$
and dynamics of a related self-map of
$2^{\mathbb {N}}$
. Our main result is the existence and uniqueness of a pair of binary sequences, each the complement-description of the other. We also take every opportunity to make puns.
For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G.
Recently, Mandache proved a “popular differences” version of this result in the finite field case
$G = {\mathbb{F}}_p^n$
, showing that for any subset A ⊂ G × G of density α, one can fix d ≠ 0 such that A contains a large fraction, now known to be approximately α4, of all corners with difference d, as x, y vary over G. We generalise Mandache’s result to all compact abelian groups G, as well as the case of corners in
$\mathbb{Z}^2$
.
For a given set
$S\subseteq \mathbb {Z}_m$
and
$\overline {n}\in \mathbb {Z}_m$
, let
$R_S(\overline {n})$
denote the number of solutions of the equation
$\overline {n}=\overline {s}+\overline {s'}$
with ordered pairs
$(\overline {s},\overline {s'})\in S^2$
. We determine the structure of
$A,B\subseteq \mathbb {Z}_m$
with
$|(A\cup B)\setminus (A\cap B)|=m-2$
such that
$R_{A}(\overline {n})=R_{B}(\overline {n})$
for all
$\overline {n}\in \mathbb {Z}_m$
, where m is an even integer.
In this paper, we decompose
$\overline {D}(a,M)$
into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let
$\overline {p}(n)$
be the number of partitions of n and
$\overline {N}(a,M,n)$
be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:
We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.
We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.
Given
$d\in \mathbb{N}$
, we establish sum-product estimates for finite, nonempty subsets of
$\mathbb{R}^{d}$
. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let
$A$
be a finite, nonempty set of
$d\times d$
diagonal matrices with real entries. Then, for all
$\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$
,
which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’, Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.
Recently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.
In this paper we continue the study of automorphism groups of constant-length substitution shifts and also their topological factors. We show that, up to conjugacy, all roots of the identity map are letter-exchanging maps, and all other non-trivial automorphisms arise from twisted compressions of another constant-length substitution. We characterize the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant-length substitution shift is topologically conjugate to a constant-length substitution shift via a letter-to-letter code.
In this note we use some
$q$
-congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the
$m=5$
case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as
$z$
approaches roots of unity of degree
$k^{n}$
, where
$k$
is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over
$\mathbb{C}(z)$
. Finally, we discuss asymptotic bounds towards generic points on the unit circle.
Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.
We introduce a notion of
$q$
-deformed rational numbers and
$q$
-deformed continued fractions. A
$q$
-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the
$q$
-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the
$q$
-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties,
$q$
-deformation of the Farey graph, matrix presentations and
$q$
-continuants are given, as well as a relation to the Jones polynomial of rational knots.
We prove irregularities in the distribution of prime numbers in any Beatty sequence
${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$
, where
$\unicode[STIX]{x1D6FC}$
is a positive real irrational number of finite type.
Such a sequence is eventually periodic and we denote by
$P(n)$
the maximal period of such sequences for given odd
$n$
. We prove a lower bound for
$P(n)$
by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo
$n$
.
We show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup
$H \subset {A^{{K^{{O_s}(1)}}}}$
modulo which
${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$
contains a nilprogression of rank at most
${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$
and size at least
$\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$
. This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.
We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers
$\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$
of
$\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$
, where
$\unicode[STIX]{x1D6E5}$
is a finite-index subgroup of
$\text{SL}(n+1,\mathbb{Z})$
. These subsets can be obtained by projecting to the hyperplane
$\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$
sets of the form
$\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$
, where for all
$j$
,
$\mathbf{a}_{j}$
is a primitive lattice point in
$\mathbb{Z}^{n+1}$
. Our method involves applying the equidistribution of expanding horospheres in quotients of
$\text{SL}(n+1,\mathbb{R})$
developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in
$\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$
when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form
$\mathbf{A}$
.
For every integer
$k\geq 2$
and every
$A\subseteq \mathbb{N}$
, we define the
$k$
-directions sets of
$A$
as
$D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$
and
$D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$
, where
$\Vert \cdot \Vert$
is the Euclidean norm and
$A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$
. Via an appropriate homeomorphism,
$D^{k}(A)$
is a generalisation of the ratio set
$R(A):=\{a/b:a,b\in A\}$
. We study
$D^{k}(A)$
and
$D^{\text{}\underline{k}}(A)$
as subspaces of
$S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$
. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets
$X\subseteq S^{k-1}$
such that there exists
$A\subseteq \mathbb{N}$
satisfying
$D^{\text{}\underline{k}}(A)^{\prime }=X$
, where
$Y^{\prime }$
denotes the set of accumulation points of
$Y$
. Moreover, we provide a simple sufficient condition for
$D^{k}(A)$
to be dense in
$S^{k-1}$
. We conclude with questions for further research.
Suppose
$a^{2}(a^{2}+1)$
divides
$b^{2}(b^{2}+1)$
with
$b>a$
. We improve a previous result and prove a gap principle, without any additional assumptions, namely
$b\gg a(\log a)^{1/8}/(\log \log a)^{12}$
. We also obtain
$b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$
under the abc conjecture.