arXiv subjects

Scott Duke Kominers

Publications and source records attributed to Scott Duke Kominers.

At least 37 records · Page 2Linked to original sources

arXiv2026

How Thick Is the Sierpiński Triangle?

Although the Sierpiński triangle has planar area $0$, it is uniformly non-flat: at every point and every scale, its nearby points span a two-dimensional region of comparable size. We prove a sharp version of this statement, showing that the Feng--Wu thickness of $E$ is exactly $\sqrt{3}/6$, the inradius of a unit equilateral triangle. More precisely, if $E$ is the standard Sierpiński triangle of side length $1$ and $B(x,r)$ denotes the closed disk of radius $r$ centered at $x$, then for every $x\in E$ and every $0<r\le 1$, the convex hull of $E\cap B(x,r)$ contains an equilateral triangle of side length $r$. Consequently, $\operatorname{conv}(E\cap B(x,r))$ contains a closed disk of radius $(\sqrt{3}/6)r$; this constant is best possible. The proof is elementary -- boundary edges of all construction triangles survive in the limit set, and self-similarity reduces the problem to the normalized range $1/2\le r\le 1$.

math.MG↗

arXiv2026

A Fixed-Prime Criterion for Reciprocals in Missing-Digit Sets

We prove a structural upper bound on the $p$-adic valuation of denominators of rationals belonging to a missing-digit set $K_{m,D}$, generalizing a key step in recent work of Lin, Wu, and Yang [arXiv:2603.24614] on reciprocals of factorials. For a rational $\frac{r}{Q}$ with $\gcd(Q,m)=1$ and a fixed prime $p_0\nmid m$, membership in $K_{m,D}$ forces $ν_{p_0}(Q)$ to be controlled by the $p_0$-adic valuation of the multiplicative order of $m$ modulo the radical of $Q$, with explicit overhead depending only on $m$ and $D$. Because the obstruction is stated at the level of a single denominator, a pair of sequence-specific valuation estimates converts it into an effective finiteness criterion for $\left\{\frac{1}{a_n}:n\in\mathbb{N}\right\}\cap K_{m,D}$. Specializing to the case in which $Q$ is the part of $n!$ coprime to $m$ recovers the fixed-prime step in the Lin--Wu--Yang argument. As applications, we treat reciprocals of superfactorials, products of polynomial values, and products of Fibonacci numbers. We also exhibit an exponential family -- products of $(m^k-1)$ -- for which the full structural criterion applies but a coarser largest-prime-factor formulation does not.

math.NT↗

arXiv2026

An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0 6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

math.MG↗

arXiv2026

Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to $b$ contains infinitely many integers that are simultaneously $b$-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell$-Niven-ness for $\ell=1,\ldots, k$).

math.NT↗

arXiv2025

Shill-Proof Auctions

We characterize single-item auction formats that are shill-proof in the sense that a profit-maximizing seller has no incentive to submit shill bids. We distinguish between strong shill-proofness, in which a seller with full knowledge of bidders' valuations can never profit from shilling, and weak shill-proofness, which requires only that the expected equilibrium profit from shilling is non-positive. The Dutch auction (with a suitable reserve) is the unique (revenue-)optimal and strongly shill-proof auction. Any deterministic auction can satisfy only two properties in the set {static, strategy-proof, weakly shill-proof}. Our main results extend to settings with affiliated and interdependent values.

econ.TH↗

arXiv2025

A Flanking Pattern in a Sum-of-Divisors Congruence

We consider composite $n$ satisfying the congruence $$n \cdot σ_k(n) \equiv 2 \pmod{ϕ(n)},$$ and show a "flanking" structure: $14$ appears in both $S_{k-1}$ and $S_{k+1}$ whenever certain values of $n$ appear in $S_k$; and, moreover, $14$ is the only (nontrivial) case of this property. Along the way, we derive a new characterization of the $n$ that appear in the sets $S_{k}$.

math.NT↗

arXiv2025

Uniform Bounds for Digit-Appending Fibonacci Walks

Building on the work of Miller et al. [Fibonacci Quarterly, 2022], we show that it is impossible to "walk to infinity" along the Fibonacci sequence in any integer base $b\geq 2$ when at most $N$ digits are appended per step. Our proof method is base-independent, yielding the bound \[L \;\leq\; 2N\log_φb \,+\, O(1),\] uniformly in the starting term, without relying on base-specific periodicity computations (here, $φ=\frac{1+\sqrt{5}}{2}$). Our approach extends to certain Lucas sequences.

math.NT↗

arXiv2025

Rationally Analyzing Shelby: Proving Incentive Compatibility in a Decentralized Storage Network

Decentralized storage is one of the most natural applications built on blockchains and a central component of the Web3 ecosystem. Yet despite a decade of active development -- from IPFS and Filecoin to more recent entrants -- most of these storage protocols have received limited formal analysis of their incentive properties. Claims of incentive compatibility are sometimes made, but rarely proven. This gap matters: without well-designed incentives, a system may distribute storage but fail to truly decentralize it. We analyze Shelby -- a storage network protocol recently proposed by Aptos Labs and Jump Crypto -- and provide the first formal proof of its incentive properties. Our game-theoretic model shows that while off-chain audits alone collapse to universal shirking, Shelby's combination of peer audits with occasional on-chain verification yields incentive compatibility under natural parameter settings. We also examine coalition behavior and outline a simple modification that strengthens the protocol's collusion-resilience.

cs.GT↗

arXiv2025

NFTs as a Data-Rich Test Bed: Conspicuous Consumption and its Determinants

Conspicuous consumption occurs when a consumer derives value from a good based on its social meaning as a signal of wealth, taste, and/or community affiliation. Common conspicuous goods include designer footwear, country club memberships, and artwork; conspicuous goods also exist in the digital sphere, with non-fungible tokens (NFTs) as a prominent example. The NFT market merits deeper study for two key reasons: first, it is poorly understood relative to its economic scale; and second, it is unusually amenable to analysis because NFT transactions are publicly available on the blockchain, making them useful as a test bed for conspicuous consumption dynamics. This paper introduces a model that incorporates two previously identified elements of conspicuous consumption: the \emph{bandwagon effect} (goods increase in value as they become more popular) and the \emph{snob effect} (goods increase in value as they become rarer). Our model resolves the apparent tension between these two effects, exhibiting net complementarity between others' and one's own conspicuous consumption. We also introduce a novel dataset combining NFT transactions with embeddings of the corresponding NFT images computed using an off-the-shelf vision transformer architecture. We use our dataset to validate the model, showing that the bandwagon effect raises an NFT collection's value as more consumers join, while the snob effect drives consumers to seek rarer NFTs within a given collection.

cs.CY↗

arXiv2025

Incentive-Compatible Recovery from Manipulated Signals, with Applications to Decentralized Physical Infrastructure

We introduce the first formal model capturing the elicitation of unverifiable information from a party (the "source") with implicit signals derived by other players (the "observers"). Our model is motivated in part by applications in decentralized physical infrastructure networks (a.k.a. "DePIN"), an emerging application domain in which physical services (e.g., sensor information, bandwidth, or energy) are provided at least in part by untrusted and self-interested parties. A key challenge in these signal network applications is verifying the level of service that was actually provided by network participants. We first establish a condition called source identifiability, which we show is necessary for the existence of a mechanism for which truthful signal reporting is a strict equilibrium. For a converse, we build on techniques from peer prediction to show that in every signal network that satisfies the source identifiability condition, there is in fact a strictly truthful mechanism, where truthful signal reporting gives strictly higher total expected payoff than any less informative equilibrium. We furthermore show that this truthful equilibrium is in fact the unique equilibrium of the mechanism if there is positive probability that any one observer is unconditionally honest (e.g., if an observer were run by the network owner). Also, by extending our condition to coalitions, we show that there are generally no collusion-resistant mechanisms in the settings that we consider. We apply our framework and results to two DePIN applications: proving location, and proving bandwidth. In the location-proving setting observers learn (potentially enlarged) Euclidean distances to the source. Here, our condition has an appealing geometric interpretation, implying that the source's location can be truthfully elicited if and only if it is guaranteed to lie inside the convex hull of the observers.

cs.GT↗

arXiv2024

A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves

In the modular robot reconfiguration problem, we are given $n$ cube-shaped modules (or robots) as well as two configurations, i.e., placements of the $n$ modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times. For many years it has been known that certain module configurations in this model require at least $Ω(n^2)$ moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm -- i.e., we show that any $n$-module configuration can reconfigure itself into any specified $n$-module configuration using just sliding moves. Our algorithm achieves reconfiguration in $O(n^2)$ moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.

cs.CG↗

arXiv2023

To Infinity and Beyond: A General Framework for Scaling Economic Theories

Many economic theory models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. We provide a principled framework for scaling results from such models by removing these finiteness assumptions. Our sufficient conditions are on the theorem statement only, and not on its proof. This results in short proofs, and even allows to use the same argument to scale similar theorems that were proven using distinctly different tools. We demonstrate the versatility of our approach via examples from both revealed-preference theory and matching theory.

cs.GT↗

arXiv2022

The Harvard USPTO Patent Dataset: A Large-Scale, Well-Structured, and Multi-Purpose Corpus of Patent Applications

Innovation is a major driver of economic and social development, and information about many kinds of innovation is embedded in semi-structured data from patents and patent applications. Although the impact and novelty of innovations expressed in patent data are difficult to measure through traditional means, ML offers a promising set of techniques for evaluating novelty, summarizing contributions, and embedding semantics. In this paper, we introduce the Harvard USPTO Patent Dataset (HUPD), a large-scale, well-structured, and multi-purpose corpus of English-language patent applications filed to the United States Patent and Trademark Office (USPTO) between 2004 and 2018. With more than 4.5 million patent documents, HUPD is two to three times larger than comparable corpora. Unlike previously proposed patent datasets in NLP, HUPD contains the inventor-submitted versions of patent applications--not the final versions of granted patents--thereby allowing us to study patentability at the time of filing using NLP methods for the first time. It is also novel in its inclusion of rich structured metadata alongside the text of patent filings: By providing each application's metadata along with all of its text fields, the dataset enables researchers to perform new sets of NLP tasks that leverage variation in structured covariates. As a case study on the types of research HUPD makes possible, we introduce a new task to the NLP community--namely, binary classification of patent decisions. We additionally show the structured metadata provided in the dataset enables us to conduct explicit studies of concept shifts for this task. Finally, we demonstrate how HUPD can be used for three additional tasks: multi-class classification of patent subject areas, language modeling, and summarization.

cs.CL↗

arXiv2021

Recommending with Recommendations

Recommendation systems are a key modern application of machine learning, but they have the downside that they often draw upon sensitive user information in making their predictions. We show how to address this deficiency by basing a service's recommendation engine upon recommendations from other existing services, which contain no sensitive information by nature. Specifically, we introduce a contextual multi-armed bandit recommendation framework where the agent has access to recommendations for other services. In our setting, the user's (potentially sensitive) information belongs to a high-dimensional latent space, and the ideal recommendations for the source and target tasks (which are non-sensitive) are given by unknown linear transformations of the user information. So long as the tasks rely on similar segments of the user information, we can decompose the target recommendation problem into systematic components that can be derived from the source recommendations, and idiosyncratic components that are user-specific and cannot be derived from the source, but have significantly lower dimensionality. We propose an explore-then-refine approach to learning and utilizing this decomposition; then using ideas from perturbation theory and statistical concentration of measure, we prove our algorithm achieves regret comparable to a strong skyline that has full knowledge of the source and target transformations. We also consider a generalization of our algorithm to a model with many simultaneous targets and no source. Our methods obtain superior empirical results on synthetic benchmarks.

cs.LG↗

arXiv2020

Prisoners, Rooms, and Lightswitches

We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large number of times. The rooms are indistinguishable, except that each one has $s$ lightswitches; the prisoners win their freedom if at some point a prisoner can correctly declare that each prisoner has been in every room at least once. What is the minimum number of switches per room, $s$, such that the prisoners can manage this? We show that if the prisoners do not know the switches' starting configuration, then they have no chance of escape -- but if the prisoners do know the starting configuration, then the minimum sufficient $s$ is surprisingly small. The analysis gives rise to a number of puzzling open questions, as well.

cs.DC↗

arXiv2020

Generalization by Recognizing Confusion

A recently-proposed technique called self-adaptive training augments modern neural networks by allowing them to adjust training labels on the fly, to avoid overfitting to samples that may be mislabeled or otherwise non-representative. By combining the self-adaptive objective with mixup, we further improve the accuracy of self-adaptive models for image recognition; the resulting classifier obtains state-of-the-art accuracies on datasets corrupted with label noise. Robustness to label noise implies a lower generalization gap; thus, our approach also leads to improved generalizability. We find evidence that the Rademacher complexity of these algorithms is low, suggesting a new path towards provable generalization for this type of deep learning model. Last, we highlight a novel connection between difficulties accounting for rare classes and robustness under noise, as rare classes are in a sense indistinguishable from label noise. Our code can be found at https://github.com/Tuxianeer/generalizationconfusion.

cs.LG↗

arXiv2020

Smarter Parking: Using AI to Identify Parking Inefficiencies in Vancouver

On-street parking is convenient, but has many disadvantages: on-street spots come at the expense of other road uses such as traffic lanes, transit lanes, bike lanes, or parklets; drivers looking for parking contribute substantially to traffic congestion and hence to greenhouse gas emissions; safety is reduced both due to the fact that drivers looking for spots are more distracted than other road users and that people exiting parked cars pose a risk to cyclists. These social costs may not be worth paying when off-street parking lots are nearby and have surplus capacity. To see where this might be true in downtown Vancouver, we used artificial intelligence techniques to estimate the amount of time it would take drivers to both park on and off street for destinations throughout the city. For on-street parking, we developed (1) a deep-learning model of block-by-block parking availability based on data from parking meters and audits and (2) a computational simulation of drivers searching for an on-street spot. For off-street parking, we developed a computational simulation of the time it would take drivers drive from their original destination to the nearest city-owned off-street lot and then to queue for a spot based on traffic and lot occupancy data. Finally, in both cases we also computed the time it would take the driver to walk from their parking spot to their original destination. We compared these time estimates for destinations in each block of Vancouver's downtown core and each hour of the day. We found many areas where off street would actually save drivers time over searching the streets for a spot, and many more where the time cost for parking off street was small. The identification of such areas provides an opportunity for the city to repurpose valuable curbside space for community-friendly uses more in line with its transportation goals.

cs.CY↗

arXiv2019

Ridesharing with Driver Location Preferences

We study revenue-optimal pricing and driver compensation in ridesharing platforms when drivers have heterogeneous preferences over locations. If a platform ignores drivers' location preferences, it may make inefficient trip dispatches; moreover, drivers may strategize so as to route towards their preferred locations. In a model with stationary and continuous demand and supply, we present a mechanism that incentivizes drivers to both (i) report their location preferences truthfully and (ii) always provide service. In settings with unconstrained driver supply or symmetric demand patterns, our mechanism achieves (full-information) first-best revenue. Under supply constraints and unbalanced demand, we show via simulation that our mechanism improves over existing mechanisms and has performance close to the first-best.

cs.MA↗