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We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\\subseteq \\mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\\geqslant 2n+1$ , then any dominant rational mapping $f:X{\\dashrightarrow}\\mathbf{P}^{n}$ must have degree at least $d-1$ . We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.

Let$\\unicode[STIX]{x1D703}$be an irrational real number. The map$T_{\\unicode[STIX]{x1D703}}:y\\mapsto (y+\\unicode[STIX]{x1D703})\\!\\hspace{0.6em}{\\rm mod}\\hspace{0.2em}1$from the unit interval$\\mathbf{I}= [\\!0,1\\![$(endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys. 16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map$[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$and the unilateral dyadic Bernoulli shift when$\\unicode[STIX]{x1D703}$is extremely well approximated by the rational numbers, namely, if$$\\begin{eqnarray}ınf _{q\\geq 1}q^{4}4^{q^{2}}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156 (2002), 79–101] showed that for every irrational number, the measure-preserving map$[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on$\\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map$[T_{\\unicode[STIX]{x1D703}},\\text{Id}]$and the unilateral dyadic Bernoulli shift whenever$$\\begin{eqnarray}ınf _{q\\geq 1}q^{4}~\\text{dist}(\\unicode[STIX]{x1D703},q^{-1}\\mathbb{Z})=0.\\end{eqnarray}$$This condition can be relaxed again into$$\\begin{eqnarray}ınf _{n\\geq 1}q_{n}^{3}~(a_{1}+\\cdots +a_{n})~|q_{n}\\unicode[STIX]{x1D703}-p_{n}|<+ınfty ,\\end{eqnarray}$$where$[0;a_{1},a_{2},\\ldots ]$is the continued fraction expansion and$(p_{n}/q_{n})_{n\\geq 0}$the sequence of convergents of$\\Vert \\unicode[STIX]{x1D703}\\Vert :=\\text{dist}(\\unicode[STIX]{x1D703},\\mathbb{Z})$. Whether Parry’s map is an isomorphism for every$\\unicode[STIX]{x1D703}$or not is still an open question, although we expect a positive answer.

An important empirical literature evaluates whether voters are rational by examining how electoral outcomes respond to events outside the control of politicians, such as natural disasters or economic shocks. The argument is that rational voters should not base electoral decisions on such events, so evidence that these events affect electoral outcomes is evidence of voter irrationality. We show that such events can affect electoral outcomes, even if voters are rational and have instrumental preferences. The reason is that these events change voters' opportunities to learn new information about incumbents. Thus, identifying voter (ir)rationality requires more than just identifying the impact of exogenous shocks on electoral fortunes. Our analysis highlights systematic ways in which electoral fortunes are expected to change in response to events outside incumbents' control. Such results can inform empirical work attempting to identify voter (ir)rationality.

Gaslighting—a type of psychological abuse aimed at making victims seem or feel “crazy,” creating a “surreal” interpersonal environment—has captured public attention. Despite the popularity of the term, sociologists have ignored gaslighting, leaving it to be theorized by psychologists. However, this article argues that gaslighting is primarily a sociological rather than a psychological phenomenon. Gaslighting should be understood as rooted in social inequalities, including gender, and executed in power-laden intimate relationships. The theory developed here argues that gaslighting is consequential when perpetrators mobilize genderbased stereotypes and structural and institutional inequalities against victims to manipulate their realities. Using domestic violence as a strategic case study to identify the mechanisms via which gaslighting operates, I reveal how abusers mobilize gendered stereotypes; structural vulnerabilities related to race, nationality, and sexuality; and institutional inequalities against victims to erode their realities. These tactics are gendered in that they rely on the association of femininity with irrationality. Gaslighting offers an opportunity for sociologists to theorize under-recognized, gendered forms of power and their mobilization in interpersonal relationships.

Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of \\(\\zeta(5)\\). But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

Rationality specializes in families of surfaces, even with mild singularities. In this paper, we study the analogous question for the degree of irrationality. We prove a specialization result when the degree of irrationality on the generic fiber arises from the quotient by a group action.

We prove the irrationality of the classical Dirichlet L-value \\(L(2,\\chi_{-3})\\). The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our work also establishes the \\(\\mathbf{Q}\\)-linear independence of \\(1\\), \\(\\zeta(2)\\), and \\(L(2,\\chi_{-3})\\). We also give a number of other applications of our method to other problems in irrationality.