On Alweiss's example for multiple recurrence
In this expository note we discuss the example of R.~Alweiss giving a construction of a set $S \subset \mathbf{N}$ which intersects every nil-Bohr set, but which is not a set of $2$-recurrence.
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arXiv subjects
Ben Green
Publications and source records attributed to Ben Green.
The proportion of permutations fixing a $k$-set
Denote by $p(k)$ the limit, as $n \rightarrow \infty$, of the probability that a random permutation on a set of size $n$ has an invariant set of size $k$. We give an asymptotic formula for $p(k)$, showing that it is asymptotically $f(\{\log_2 k\}) k^{-\delta} (\log k)^{-3/2}$ where $\delta = 1 - \frac{1 + \log \log 2}{\log 2} \approx 0.086$ and $f$ is a smooth, positive, function on $\mathbb{R}/\mathbb{Z}$, which we will describe explicitly. The function $f$ satisfies $\frac{\max f}{\min f} < 1 + 2 \times 10^{-7}$ and we conjecture that it is not constant. Estimating $p(k)$ is a model for the more well-known question which asks for an estimation of $M(n)$, the number of distinct elements in the $n$-by-$n$ multiplication table. By elaborating on the techniques in this paper, we will give an asymptotic for $M(n)$ in forthcoming work.
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Remarks on the inverse Littlewood conjecture
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.
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Bounds for monochromatic solutions to $\{x+y,xy\}$
Let $r$ be a sufficiently large positive integer, and let $N \ge \exp\exp(r^{50})$. Then any $r$-colouring of $[N]$ contains a monochromatic copy of $\{x+y,xy\}$ with $x > y > 2$.
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New bounds for the Furstenberg-S\'ark\"ozy theorem
Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-c\sqrt{\log N}}$.
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Primes of the form $p^2 + nq^2$
Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian primes conjecture' of Friedlander and Iwaniec. We study the problem using the method of Type I/II sums in the number field $\mathbf{Q}(\sqrt{-n})$. The main innovation is in the treatment of the Type II sums, where we make heavy use of two recent developments in the theory of Gowers norms in additive combinatorics: quantitative versions of so-called concatenation theorems, due to Kuca and to Kuca--Kravitz-Leng, and the quasipolynomial inverse theorem of Leng, Sah and the second author.
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Marton's Conjecture in abelian groups with bounded torsion
We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let $G$ be an abelian group of torsion $m$ (meaning $mg=0$ for all $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \leq K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors.
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Covering integers by $x^2 + dy^2$
What proportion of integers $n \leqslant N$ may be expressed as $x^2 + dy^2$ for some $d \leqslant \Delta$, with $x,y $ integers? Writing $\Delta$ as $(\log N)^{\log 2} 2^{\alpha \sqrt{\log \log N}}$ for some $\alpha \in (-\infty, \infty)$, we show that the answer is $\Phi(\alpha) + o(1)$, where $\Phi$ is the Gaussian distribution function $\Phi(\alpha) = \frac{1}{2\pi} \int^{\alpha}_{-\infty} e^{-x^2/2} dx$. A consequence of this is a phase transition: almost none of the integers $n \leqslant N$ can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 - \varepsilon}$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leqslant (\log N)^{\log 2 + \varepsilon}$.
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On a conjecture of Marton
We prove a conjecture of K. Marton, widely known as the polynomial Freiman--Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a subsequent paper.
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Waring's problem with restricted digits
Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then every sufficiently large integer is a sum of at most $b^{160 k^2}$ numbers of the form $x^k$, $x \in \mathcal{S}$.
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Sumsets and entropy revisited
The entropic doubling $\sigma_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $\sigma[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.
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On S\'ark\"ozy's theorem for shifted primes
Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.
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"If it didn't happen, why would I change my decision?": How Judges Respond to Counterfactual Explanations for the Public Safety Assessment
Many researchers and policymakers have expressed excitement about algorithmic explanations enabling more fair and responsible decision-making. However, recent experimental studies have found that explanations do not always improve human use of algorithmic advice. In this study, we shed light on how people interpret and respond to counterfactual explanations (CFEs) -- explanations that show how a model's output would change with marginal changes to its input(s) -- in the context of pretrial risk assessment instruments (PRAIs). We ran think-aloud trials with eight sitting U.S. state court judges, providing them with recommendations from a PRAI that includes CFEs. We found that the CFEs did not alter the judges' decisions. At first, judges misinterpreted the counterfactuals as real -- rather than hypothetical -- changes to defendants. Once judges understood what the counterfactuals meant, they ignored them, stating their role is only to make decisions regarding the actual defendant in question. The judges also expressed a mix of reasons for ignoring or following the advice of the PRAI without CFEs. These results add to the literature detailing the unexpected ways in which people respond to algorithms and explanations. They also highlight new challenges associated with improving human-algorithm collaborations through explanations.
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Technology Ethics in Action: Critical and Interdisciplinary Perspectives
This special issue interrogates the meaning and impacts of "tech ethics": the embedding of ethics into digital technology research, development, use, and governance. In response to concerns about the social harms associated with digital technologies, many individuals and institutions have articulated the need for a greater emphasis on ethics in digital technology. Yet as more groups embrace the concept of ethics, critical discourses have emerged questioning whose ethics are being centered, whether "ethics" is the appropriate frame for improving technology, and what it means to develop "ethical" technology in practice. This interdisciplinary issue takes up these questions, interrogating the relationships among ethics, technology, and society in action. This special issue engages with the normative and contested notions of ethics itself, how ethics has been integrated with technology across domains, and potential paths forward to support more just and egalitarian technology. Rather than starting from philosophical theories, the authors in this issue orient their articles around the real-world discourses and impacts of tech ethics--i.e., tech ethics in action.
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The Flaws of Policies Requiring Human Oversight of Government Algorithms
As algorithms become an influential component of government decision-making around the world, policymakers have debated how governments can attain the benefits of algorithms while preventing the harms of algorithms. One mechanism that has become a centerpiece of global efforts to regulate government algorithms is to require human oversight of algorithmic decisions. Despite the widespread turn to human oversight, these policies rest on an uninterrogated assumption: that people are able to effectively oversee algorithmic decision-making. In this article, I survey 41 policies that prescribe human oversight of government algorithms and find that they suffer from two significant flaws. First, evidence suggests that people are unable to perform the desired oversight functions. Second, as a result of the first flaw, human oversight policies legitimize government uses of faulty and controversial algorithms without addressing the fundamental issues with these tools. Thus, rather than protect against the potential harms of algorithmic decision-making in government, human oversight policies provide a false sense of security in adopting algorithms and enable vendors and agencies to shirk accountability for algorithmic harms. In light of these flaws, I propose a shift from human oversight to institutional oversight as the central mechanism for regulating government algorithms. This institutional approach operates in two stages. First, agencies must justify that it is appropriate to incorporate an algorithm into decision-making and that any proposed forms of human oversight are supported by empirical evidence. Second, these justifications must receive democratic public review and approval before the agency can adopt the algorithm.
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Quadratic forms in 8 prime variables
We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\operatorname{Sp}_8(\mathbf{Z}/q\mathbf{Z})$. Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolutions of measures, which reduces matters to understanding the action of certain 12-dimensional subgroups in the Weil representation. Sufficient understanding can be gained by using the basic represention theory of $\operatorname{SL}_2(k)$, $k$ a finite field.
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Escaping the Impossibility of Fairness: From Formal to Substantive Algorithmic Fairness
Efforts to promote equitable public policy with algorithms appear to be fundamentally constrained by the "impossibility of fairness" (an incompatibility between mathematical definitions of fairness). This technical limitation raises a central question about algorithmic fairness: How can computer scientists and policymakers support equitable policy reforms with algorithms? In this article, I argue that promoting justice with algorithms requires reforming the methodology of algorithmic fairness. First, I diagnose the problems of the current methodology for algorithmic fairness, which I call "formal algorithmic fairness." Because formal algorithmic fairness restricts analysis to isolated decision-making procedures, it leads to the impossibility of fairness and to models that exacerbate oppression despite appearing "fair." Second, I draw on theories of substantive equality from law and philosophy to propose an alternative methodology, which I call "substantive algorithmic fairness." Because substantive algorithmic fairness takes a more expansive scope of analysis, it enables an escape from the impossibility of fairness and provides a rigorous guide for alleviating injustice with algorithms. In sum, substantive algorithmic fairness presents a new direction for algorithmic fairness: away from formal mathematical models of "fair" decision-making and toward substantive evaluations of whether and how algorithms can promote justice in practice.
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The Contestation of Tech Ethics: A Sociotechnical Approach to Technology Ethics in Practice
This article introduces the special issue "Technology Ethics in Action: Critical and Interdisciplinary Perspectives". In response to recent controversies about the harms of digital technology, discourses and practices of "tech ethics" have proliferated across the tech industry, academia, civil society, and government. Yet despite the seeming promise of ethics, tech ethics in practice suffers from several significant limitations: tech ethics is vague and toothless, has a myopic focus on individual engineers and technology design, and is subsumed into corporate logics and incentives. These limitations suggest that tech ethics enables corporate "ethics-washing": embracing the language of ethics to defuse criticism and resist government regulation, without committing to ethical behavior. Given these dynamics, I describe tech ethics as a terrain of contestation where the central debate is not whether ethics is desirable, but what "ethics" entails and who gets to define it. Current approaches to tech ethics are poised to enable technologists and technology companies to label themselves as "ethical" without substantively altering their practices. Thus, those striving for structural improvements in digital technologies must be mindful of the gap between ethics as a mode of normative inquiry and ethics as a practical endeavor. In order to better evaluate the opportunities and limits of tech ethics, I propose a sociotechnical approach that analyzes tech ethics in light of who defines it and what impacts it generates in practice.
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