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We prove that if
$A \subseteq [X,\,2X]$
and
$B \subseteq [Y,\,2Y]$
are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then
$|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$
. This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
Suppose that N is 2-coloured. Then there are infinitely many monochromatic solutions to $x+y=z^{2}$. On the other hand, there is a 3-colouring of N with only finitely many monochromatic solutions to this equation.
Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).
Hip hop group Wu-Tang Clan sold only one, expensive copy of their album, Once Upon A Time in Shaolin. This exemplifies recent strategies by popular music artists to establish their work as art, with what Walter Benjamin calls ‘aura’, in response to the accessibility and dematerialisation enabled by digital technology as well as longstanding cultural condescension. Critics argue that popular music should not be restricted but shared, with digital technology increasing opportunities for shared consumption. This article considers the fate of music's aura in the age of mechanical reproduction, arguing that it does not disappear but is dispersed and diversified. The digital acceleration of mass reproduction has drawn mixed responses from artists, fans and commentators, and Shaolin and similar projects show how the separation of music from its physical commodity form has brought renewed attention to perennial tensions between the popular, artistic and commercial aspects of popular music.
Define
$r_{4}(N)$
to be the largest cardinality of a set
$A\subset \{1,\ldots ,N\}$
that does not contain four elements in arithmetic progression. In 1998, Gowers proved that
Let G be an abelian group of cardinality n, where hcf(n, 6) = 1, and let A be a random subset of G. Form a graph ΓA on vertex set G by joining x to y if and only if x + y ∈ A. Then, with high probability as n → ∞, the chromatic number χ(ΓA) is at most $(1 + o(1))\tfrac{n}{2\log_2 n}$. This is asymptotically sharp when G = ℤ/nℤ, n prime.
As many as 10% of the population experience post-traumatic stress disorder (PTSD) at some time in their lives. It often runs a severe, chronic and treatment-resistant course. This article reviews the evidence base for typically recommended treatments such as cognitive-behavioural therapy (CBT), eye movement desensitisation and reprocessing and selective serotonin reuptake inhibitors (SSRIs). It tabulates the major randomised controlled trials of SSRIs and trauma-focused CBT and reviews research on novel treatments such as ketamine, MDMA, quetiapine, propranolol and prazosin.
A survey of the Milky Way disk and the Magellanic System at the wavelengths of the 21-cm atomic hydrogen (H i) line and three 18-cm lines of the OH molecule will be carried out with the Australian Square Kilometre Array Pathfinder telescope. The survey will study the distribution of H i emission and absorption with unprecedented angular and velocity resolution, as well as molecular line thermal emission, absorption, and maser lines. The area to be covered includes the Galactic plane (|b| < 10°) at all declinations south of δ = +40°, spanning longitudes 167° through 360°to 79° at b = 0°, plus the entire area of the Magellanic Stream and Clouds, a total of 13 020 deg2. The brightness temperature sensitivity will be very good, typically σT≃ 1 K at resolution 30 arcsec and 1 km s−1. The survey has a wide spectrum of scientific goals, from studies of galaxy evolution to star formation, with particular contributions to understanding stellar wind kinematics, the thermal phases of the interstellar medium, the interaction between gas in the disk and halo, and the dynamical and thermal states of gas at various positions along the Magellanic Stream.
Any function F: {0,. . ., N − 1} → {−1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Möbius function μ in the sense that
\[
\frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) → 0 \quad\text{as}~~ N → \infty.
\]
The proof combines a result of Linial, Mansour and Nisan with techniques of Kátai and Harman, used in their work on finding primes with specified digits.
The target adopted by world leaders of significantly reducing the rate of biodiversity loss by 2010 was not met but this stimulated a new suite of biodiversity targets for 2020 adopted by the Parties to the Convention on Biological Diversity (CBD) in October 2010. Indicators will be essential for monitoring progress towards these targets and the CBD will be defining a suite of relevant indicators, building on those developed for the 2010 target. Here we argue that explicitly linked sets of indicators offer a more useful framework than do individual indicators because the former are easier to understand, communicate and interpret to guide policy. A Response-Pressure-State-Benefit framework for structuring and linking indicators facilitates an understanding of the relationships between policy actions, anthropogenic threats, the status of biodiversity and the benefits that people derive from it. Such an approach is appropriate at global, regional, national and local scales but for many systems it is easier to demonstrate causal linkages and use them to aid decision making at national and local scales. We outline examples of linked indicator sets for humid tropical forests and marine fisheries as illustrations of the concept and conclude that much work remains to be done in developing both the indicators and the causal links between them.
We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this.
By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes.
We describe the structure of ‘K-approximate subgroups’ of torsion-free nilpotent groups, paying particular attention to Lie groups.
Three other works, by Fisher et al., by Sanders and by Tao, have appeared that independently address related issues. We comment briefly on some of the connections between these papers.
We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freĭman type are equivalent to the known inverse results for the Gowers U3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces 2n, and of the cyclic groups ℤ/Nℤ.
In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in : if is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size ; except for the error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.
Let $p$ be a prime, and let $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{R}$ be a function with
$\mathbb{E}f=0$ and $||\hat{f}|{{|}_{1}}\le 1$. Then ${{\min }_{x\in \mathbb{Z}/p\mathbb{Z}}}|f\left( x \right)|=O{{\left( \log p \right)}^{-1/3+\in }}$. One should think of $f$ as being “approximately continuous”; our result is then an “approximate intermediate value theorem”.
As an immediate consequence we show that if $A\subseteq \mathbb{Z}/p\mathbb{Z}$ is a set of cardinality $\left\lfloor {p}/{2}\; \right\rfloor $, then ${{\sum }_{r}}\widehat{|\,{{1}_{A}}}\left( r \right)|\gg {{\left( \log p \right)}^{1/3-\in }}$. This gives a result on a “$\,\bmod \,p$” analogue of Littlewood's well-known problem concerning the smallest possible ${{L}^{1}}$-norm of the Fourier transform of a set of $n$ integers.
Another application is to answer a question of Gowers. If $A\,\subseteq \,{\mathbb{Z}}/{p\mathbb{Z}}\;$ is a set of size $\left\lfloor {p}/{2}\; \right\rfloor $, then there is some $x\,\in \,\mathbb{Z}/p\mathbb{Z}$ such that
$$||A\cap \left( A+x \right)\,-\,p/4|\,=o\left( p \right).$$