Large values of the Gowers-Host-Kra seminorms

The \emph{Gowers uniformity norms} $\|f\|_{U^k(G)}$ of a function f: G \to \C on a finite additive group $G$, together with the slight variant $\|f\|_{U^k([N])}$ defined for functions on a discrete interval $[N] := \{1,...,N\}$, are of importance in the modern theory of counting additive patterns (such as arithmetic progressions) inside large sets. Closely related to these norms are the \emph{Gowers-Host-Kra seminorms} $\|f\|_{U^k(X)}$ of a measurable function f: X \to \C on a measure-preserving system $X = (X, {\mathcal X}, \mu, T)$. Much recent effort has been devoted to the question of obtaining necessary and sufficient conditions for these Gowers norms to have non-trivial size (e.g. at least $\eta$ for some small $\eta > 0$), leading in particular to the inverse conjecture for the Gowers norms, and to the Host-Kra classification of characteristic factors for the Gowers-Host-Kra seminorms. In this paper we investigate the near-extremal (or "property testing") version of this question, when the Gowers norm or Gowers-Host-Kra seminorm of a function is almost as large as it can be subject to an $L^\infty$ or $L^p$ bound on its magnitude. Our main results assert, roughly speaking, that this occurs if and only if $f$ behaves like a polynomial phase, possibly localised to a subgroup of the domain; this can be viewed as a higher-order analogue of classical results of Russo and Fournier, and are also related to the polynomiality testing results over finite fields of Blum-Luby-Rubinfeld and Alon-Kaufman-Krivelevich-Litsyn-Ron. We investigate the situation further for the $U^3$ norms, which are associated to 2-step nilsequences, and find that there is a threshold behaviour, in that non-trivial 2-step nilsequences (not associated with linear or quadratic phases) only emerge once the $U^3$ norm is at most $2^{-1/8}$ of the $L^\infty$ norm.Comment: 52 pages, no figures, to appear, Journal d'Analyse Jerusalem. This is the final version, incorporating the referee's suggestion

The Logic of Conditional Belief

The logic of indicative conditionals remains the topic of deep and intractable philosophical disagreement. I show that two influential epistemic norms—the Lockean theory of belief and the Ramsey test for conditional belief—are jointly sufficient to ground a powerful new argument for a particular conception of the logic of indicative conditionals. Specifically, the argument demonstrates, contrary to the received historical narrative, that there is a real sense in which Stalnaker’s semantics for the indicative did succeed in capturing the logic of the Ramseyan indicative conditional

Polynomial Threshold Functions, AC^0 Functions and Spectral Norms

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_∞^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived

The Relationship Between Belief and Credence

Sometimes epistemologists theorize about belief, a tripartite attitude on which one can believe, withhold belief, or disbelieve a proposition. In other cases, epistemologists theorize about credence, a fine-grained attitude that represents one’s subjective probability or confidence level toward a proposition. How do these two attitudes relate to each other? This article explores the relationship between belief and credence in two categories: descriptive and normative. It then explains the broader significance of the belief-credence connection and concludes with general lessons from the debate thus far

Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding

Denoising by frame thresholding is one of the most basic and efficient methods for recovering a discrete signal or image from data that are corrupted by additive Gaussian white noise. The basic idea is to select a frame of analyzing elements that separates the data in few large coefficients due to the signal and many small coefficients mainly due to the noise \epsilon_n. Removing all data coefficients being in magnitude below a certain threshold yields a reconstruction of the original signal. In order to properly balance the amount of noise to be removed and the relevant signal features to be kept, a precise understanding of the statistical properties of thresholding is important. For that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n} || for a wide class of redundant frames (\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give a rationale for universal extreme value thresholding techniques yielding asymptotically sharp confidence regions and smoothness estimates corresponding to prescribed significance levels. The results cover many frames used in imaging and signal recovery applications, such as redundant wavelet systems, curvelet frames, or unions of bases. We show that `generically' a standard Gumbel law results as it is known from the case of orthonormal wavelet bases. However, for specific highly redundant frames other limiting laws may occur. We indeed verify that the translation invariant wavelet transform shows a different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have slightely changed the title of the paper and we have rewritten parts of the introduction. Except for corrected typos the other parts of the paper are the same as the original versions