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We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain
$C(\log H)^{\unicode[STIX]{x1D702}}$
bounds for the number of algebraic points of height at most
$H$
on certain subsets of the graphs of such functions. The constant
$C$
and exponent
$\unicode[STIX]{x1D702}$
depend on data associated with the functions and can be effectively computed from them.
Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.
We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball
$\mathbb{B}^{n}$
using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice
$\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$
and the associated uniformization map
$\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$
. Given an algebraic subset
$S\,\subset \,\mathbb{B}^{n}$
and writing
$Z$
for the Zariski closure of
$\unicode[STIX]{x1D70B}(S)$
in
$X_{\unicode[STIX]{x1D6E4}}$
(which is equipped with a canonical quasi-projective structure), in some precise sense we realize
$Z$
as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component
$\widetilde{Z}$
of
$\unicode[STIX]{x1D70B}^{-1}(Z)$
as
$\widetilde{Z}$
exits the boundary
$\unicode[STIX]{x2202}\mathbb{B}^{n}$
by exploiting the strict pseudoconvexity of
$\mathbb{B}^{n}$
, culminating in the proof that
$\widetilde{Z}\,\subset \,\mathbb{B}^{n}$
is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of
$\text{ Aut}(\unicode[STIX]{x1D6FA})$
for (possibly reducible) bounded symmetric domains
$\unicode[STIX]{x1D6FA}$
.
In the field
$\mathbb{K}$
of formal power series over a finite field
$K$
, we consider some lacunary power series with algebraic coefficients in a finite extension of
$K(x)$
. We show that the values of these series at nonzero algebraic arguments in
$\mathbb{K}$
are
$U$
-numbers.
Let
$\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$
be a non-decreasing function,
$a_{n}(x)$
the
$n$
th partial quotient of
$x$
and
$q_{n}(x)$
the denominator of the
$n$
th convergent. The set of
$\unicode[STIX]{x1D6F9}$
-Dirichlet non-improvable numbers,
$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
is related with the classical set of
$1/q^{2}\unicode[STIX]{x1D6F9}(q)$
-approximable numbers
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
in the sense that
${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$
. Both of these sets enjoy the same
$s$
-dimensional Hausdorff measure criterion for
$s\in (0,1)$
. We prove that the set
$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$
is uncountable by proving that its Hausdorff dimension is the same as that for the sets
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
and
$G(\unicode[STIX]{x1D6F9})$
. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika64(2) (2018), 502–518].
We give an integrability criterion on a real-valued non-increasing function
$\unicode[STIX]{x1D713}$
guaranteeing that for almost all (or almost no) pairs
$(A,\mathbf{b})$
, where
$A$
is a real
$m\times n$
matrix and
$\mathbf{b}\in \mathbb{R}^{m}$
, the system
is solvable in
$\mathbf{p}\in \mathbb{Z}^{m}$
,
$\mathbf{q}\in \mathbb{Z}^{n}$
for all sufficiently large
$T$
. The proof consists of a reduction to a shrinking target problem on the space of grids in
$\mathbb{R}^{m+n}$
. We also comment on the homogeneous counterpart to this problem, whose
$m=n=1$
case was recently solved, but whose general case remains open.
For
$\unicode[STIX]{x1D6FC}$
an algebraic integer of any degree
$n\geqslant 2$
, it is known that the discriminants of the orders
$\mathbb{Z}[\unicode[STIX]{x1D6FC}^{k}]$
go to infinity as
$k$
goes to infinity. We give a short proof of this result.
Building upon ideas of the second and third authors, we prove that at least
$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$
values of the Riemann zeta function at odd integers between 3 and
$s$
are irrational, where
$\unicode[STIX]{x1D700}$
is any positive real number and
$s$
is large enough in terms of
$\unicode[STIX]{x1D700}$
. This lower bound is asymptotically larger than any power of
$\log s$
; it improves on the bound
$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$
that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.
The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider
$m$
runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least
$1/m$
from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.
We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].
In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products
$f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$
,
$d\in \mathbb{N}$
,
$d\geq 2$
, which generalize the generating function
$f_{2}(x)$
of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of
$x^{-d+1}f_{d}(x)$
have a regular structure. We also address the question of whether the corresponding Mahler numbers
$f_{d}(a)\in \mathbb{R}$
,
$a,d\in \mathbb{N}$
,
$a,d\geq 2$
, are badly approximable.
An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.
In this article, we prove the transcendence of certain infinite sums and products by applying the subspace theorem. In particular, we extend the results of Hančl and Rucki [‘The transcendence of certain infinite series’, Rocky Mountain J. Math.35 (2005), 531–537].
We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.
We exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic.
Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial
$p(n)$
with at least one irrational coefficient (except for the constant one) and integer
$m\geqslant 2$
, the sequence
$\lfloor p(n)\rfloor \hspace{0.2em}{\rm mod}\hspace{0.2em}m$
is never automatic.
We also prove that the conjecture is equivalent to the claim that the set of powers of an integer
$k\geqslant 2$
is not given by a generalised polynomial.
It is pointed out that the generalized Lambert series
$\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $
studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.
We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.