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Employing the inverse function theorem on Banach spaces, we prove that in a
$C^{2}(S^{n-1})$
-neighborhood of the unit ball, the only solutions of
$\unicode[STIX]{x1D6F1}^{2}K=cK$
are origin-centered ellipsoids. Here
$K$
is an
$n$
-dimensional convex body,
$\unicode[STIX]{x1D6F1}K$
is the projection body of
$K$
and
$\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$
We study the Goldbach problem for primes represented by the polynomial
$x^{2}+y^{2}+1$
. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers
$n$
satisfying certain necessary local conditions are representable as the sum of two primes of the form
$x^{2}+y^{2}+1$
. This improves a result of Matomäki, which tells us that almost all even
$n$
satisfying a local condition are the sum of one prime of the form
$x^{2}+y^{2}+1$
and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd
$n$
is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form
$x^{2}+y^{2}+1$
contain infinitely many three-term arithmetic progressions, and that the numbers
$\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$
, with
$\unicode[STIX]{x1D6FC}$
irrational and
$p$
running through primes of the form
$x^{2}+y^{2}+1$
, are distributed rather uniformly.
If
$3\leqslant n<\unicode[STIX]{x1D714}$
and
$V$
is a vector space over
$\mathbb{Q}$
with
$|V|\leqslant \aleph _{n-2}$
, then there is a well ordering of
$V$
such that every vector is the last element of only finitely many length-
$n$
arithmetic progressions (
$n$
-APs). This implies that there is a set mapping
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$
with no free set which is an
$n$
-AP. If, however,
$|V|\geqslant \aleph _{n-1}$
, then for every set mapping
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$
there is a free set which is an
$n$
-AP.
The purpose of this paper twofold. Firstly, we establish
$\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness and
$\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the
$\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness and
$\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if
$\unicode[STIX]{x1D707}$
is a self-similar measure satisfying the strong separation condition and
$\unicode[STIX]{x1D707}$
is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of
$\unicode[STIX]{x1D707}$
defined by
In 2006 Brown asked the following question in the spirit of Ramsey theory: given a non-periodic infinite word
$x=x_{1}x_{2}x_{3}\ldots$
with values in a set
$\mathbb{A}$
, does there exist a finite colouring
$\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow C$
relative to which
$x$
does not admit a
$\unicode[STIX]{x1D711}$
-monochromatic factorization, i.e. a factorization of the form
$x=u_{1}u_{2}u_{3}\ldots$
with
$\unicode[STIX]{x1D711}(u_{i})=\unicode[STIX]{x1D711}(u_{\!j})$
for all
$i,j\geqslant 1$
? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and for various classes of uniformly recurrent words including Sturmian words and fixed points of strongly recognizable primitive substitutions. In this paper we give a complete and optimal affirmative answer to this question by showing that if
$x=x_{1}x_{2}x_{3}\ldots$
is an infinite non-periodic word with values in a set
$\mathbb{A}$
, then there exists a
$2$
-colouring
$\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow \{0,1\}$
such that for any factorization
$x=u_{1}u_{2}u_{3}\ldots$
we have
$\unicode[STIX]{x1D711}(u_{i})\neq \unicode[STIX]{x1D711}(u_{\!j})$
for some
$i\neq j$
.
For a non-empty polyhedral set
$P\subset \mathbb{R}^{d}$
, let
${\mathcal{F}}(P)$
denote the set of faces of
$P$
, and let
$N(P,F)$
be the normal cone of
$P$
at the non-empty face
$F\in {\mathcal{F}}(P)$
. We prove the identity
$$\begin{eqnarray}\mathop{\sum }_{F\in {\mathcal{F}}(P)}(-1)^{\operatorname{dim}F}\unicode[STIX]{x1D7D9}_{F-N(P,F)}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }P\text{ is bounded},\\ 0\quad & \text{if }P\text{ is unbounded and line-free}.\end{array}\right.\end{eqnarray}$$
Previously, this formula was known to hold everywhere outside some exceptional set of Lebesgue measure
$0$
or for polyhedral cones. The case of a not necessarily line-free polyhedral set is also covered by our general theorem.
We investigate the approximation of quadratic Dirichlet
$L$
-functions over function fields by truncations of their Euler products. We first establish representations for such
$L$
-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an
$L$
-function away from its zeros and that, when the length of the product tends to infinity, we recover the original
$L$
-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet
$L$
-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet
$L$
-function over a function field, an auxiliary function based on the approximate functional equation that equals the
$L$
-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated
$L$
-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated
$L$
-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.
Let
$\unicode[STIX]{x1D6E4}\subseteq \operatorname{PSL}(2,\mathbf{R})$
be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the
$\unicode[STIX]{x1D6E4}$
-orbit of
$z$
in a hyperbolic circle around
$w$
of radius
$R$
, where
$z$
and
$w$
are given points of the upper half plane and
$R$
is a large number. An estimate with error term
$\text{e}^{(2/3)R}$
is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case
$\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$
taking
$z=w$
and averaging over
$z$
in a certain way the error term can be improved to
$\text{e}^{(7/12+\unicode[STIX]{x1D716})R}$
. Here we show such an improvement for a general
$\unicode[STIX]{x1D6E4}$
; our error term is
$\text{e}^{(5/8+\unicode[STIX]{x1D716})R}$
(which is better than
$\text{e}^{(2/3)R}$
but weaker than the estimate of Risager and Petridis in the case
$\unicode[STIX]{x1D6E4}=\operatorname{PSL}(2,\mathbf{Z})$
). Our main tool is our generalization of the Selberg trace formula proved earlier.
We study a kernel function of the twisted symmetric square
$L$
-function of elliptic modular forms. As an application, several exact special values of the
$L$
-function are computed.
We show that for
$p\geqslant 1$
, the
$p$
th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak
$p$
th moment provided that
$2q$
th and
$q$
th integral moments of these variables are comparable for all
$q\geqslant 2$
. The latest condition turns out to be necessary in the independent and identically distributed case.
It has been known since Vinogradov that, for irrational
$\unicode[STIX]{x1D6FC}$
, the sequence of fractional parts
$\{\unicode[STIX]{x1D6FC}p\}$
is equidistributed in
$\mathbb{R}/\mathbb{Z}$
as
$p$
ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.
We prove asymptotic formulas for the number of integers at most
$x$
that can be written as the product of
$k~({\geqslant}2)$
distinct primes
$p_{1}\cdots p_{k}$
with each prime factor in an arithmetic progression
$p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$
,
$(a_{j},q)=1$
$(q\geqslant 3,1\leqslant j\leqslant k)$
. For any
$A>0$
, our result is uniform for
$2\leqslant k\leqslant A\log \log x$
. Moreover, we show that there are large biases toward certain arithmetic progressions
$(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$
, and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.
Almost-flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non-spin, the rest are, in fact, parallelizable.
Let
$\unicode[STIX]{x1D703}$
be an irrational number and
$\unicode[STIX]{x1D711}:\mathbb{N}\rightarrow \mathbb{R}^{+}$
be a monotone decreasing function tending to zero. Let
$$\begin{eqnarray}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <\unicode[STIX]{x1D711}(n),\text{for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
i.e. the set of points which are approximated by the irrational rotation with respect to the error function
$\unicode[STIX]{x1D711}(n)$
. In this article, we give a complete description of the Hausdorff dimension of
$E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$
for any monotone function
$\unicode[STIX]{x1D711}$
and any irrational
$\unicode[STIX]{x1D703}$
.
that arose during our attempts to develop a two-weight theory for the Hilbert transform in
$L^{p}$
. Boundedness of
$T^{\unicode[STIX]{x1D70E}}$
is characterized when
$p\in [2,\infty )$
in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.