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This paper proposes an empirical model of dynamic discrete choice to allow for non-separable time preferences, generalizing the well-known Rust (1987) model. Under weak conditions, we show the existence of value functions and hence well-defined optimal choices. We construct a contraction mapping of the value function and propose an estimation method similar to Rust's nested fixed point algorithm. Finally, we apply the framework to the bus engine replacement data. We improve the fit of the data with our general model and reject the null hypothesis that Harold Zuercher has separable time preferences. Misspecifying an agent's preference as time-separable when it is not leads to biased inferences about structure parameters (such as the agent's risk attitudes) and misleading policy recommendations.

Let $\lfloor x\rfloor$ denote the integer part of $x$. In 1947, Mills constructed a real number $\xi$ greater than $1$ such that $\lfloor \xi^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $\xi$ satisfying this property. In this article, we determine that Mills' constant is irrational. Moreover, we also obtain partial results on the transcendency of the constant., Comment: 11 pages

After the work of Bordell\`{e}s, Dai, Heyman, Pan and Shparlinki (2018) and Heyman (2019), several authors studied the averages of arithmetic functions over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper, we give three remarks on this topic. Firstly, we improve the result of Wu and Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic progressions by using a variant of Dirichlet's hyperbola method. Secondly, we prove an asymptotic formula for the number of primitive lattice points with coordinates of the form $[x/n]$, for which we introduce a certain averaging trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh divisor problem. We derive asymptotic formulas for such "multiplicative" Titchmarsh divisor problems for "small" arithmetic functions and the Euler totient function with the von Mangoldt function. However, it turns out that the average of the Euler totient function over the $[x/p]$-sequence seems rather difficult and we propose a hypothetical asymptotic formula for this average., Comment: 27 pages

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on arithmetic progressions. As a consequence, we show that \[ \lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \] if and only if $2^{p/q}$ is simply normal to base $2$., Comment: 22 pages

An integral sequence of the form $\lfloor n^{\alpha}\rfloor$ $(n=1,2,\ldots)$ for some non-integral $\alpha>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(\alpha)$ denote the set of all those terms. In this article, we reveal that $x+y=z$ has at most finitely many solutions $(x,y,z)\in \mathrm{PS}(\alpha)^3$ for almost every $\alpha>3$ in the sense of one-dimensional Lebesgue measure. Furthermore, we show that $\mathrm{PS}(\alpha)$ has at most finitely many arithmetic progressions of length $3$ for almost every $\alpha>10$. In addition, we obtain results on the Hausdorff dimension and a more general equation such as $y=a_1x_1+\cdots +a_nx_n$., Comment: 20 pages

The random utility model is one of the most fundamental models in economics. Falmagne (1978) provides an axiomatization but his axioms can be applied only when choice frequencies of all alternatives from all subsets are observable. In reality, however, it is often the case that we do not observe choice frequencies of some alternatives. For such a dataset, we obtain a finite system of linear inequalities that is necessary and sufficient for the dataset to be rationalized by a random utility model. Moreover, the necessary and sufficient condition is tight in the sense that none of the inequalities is implied by the other inequalities, and dropping any one of the inequalities makes the condition not sufficient.

The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak (1960). Fishburn (1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). Apart for a recognition of the facets by Suck (2002), the geometric structure of the MCP was apparently not much investigated. Recently, Chang, Narita and Saito (2022) refer to the adjacency of vertices while Turansick (2022) uses a condition which we show to be equivalent to the non-adjacency of two vertices. We characterize the adjacency of vertices and the adjacency of facets. To derive a more enlightening proof of Falmagne Theorem and of Suck result, Fiorini (2004) assimilates the MCP with the flow polytope of some acyclic network. Our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor \alpha n^2 \rfloor$ for $n\geq \alpha^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(\alpha)$, the set of all such tuples, is infinite for all $\alpha\in (0,1)\cap \mathbb{Q}$ and for almost all $\alpha\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $\alpha>0$ such that $T(\alpha)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $\lceil \alpha n^2 \rceil$ for $n\in \mathbb{N}$., Comment: 23 pages, 2 figures

We obtain a necessary and sufficient condition under which random-coefficient discrete choice models, such as mixed-logit models, are rich enough to approximate any nonparametric random utility models arbitrarily well across choice sets. The condition turns out to be the affine-independence of the set of characteristic vectors. When the condition fails, resulting in some random utility models that cannot be closely approximated, we identify preferences and substitution patterns that are challenging to approximate accurately. We also propose algorithms to quantify the magnitude of approximation errors.

Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set. The first goal of this article is to determine the topological structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in \mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to the Cantor middle third set for some $a$. The second goal is to propose an algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidly increasing. As a corollary, we disclose that the minimum of $\mathcal{W}(k)$ is transcendental. In addition, we apply the main result to the set of $A>1$ such that the integer part of $A^{3^{k!}}$ is always a prime number. As a consequence, we give a certain infinite subset of this set which is algebraically independent. Furthermore, we also get results on the rational approximation, $\mathbb{Q}$-linear independence, and numerical calculations of elements in $\mathcal{W}(c_k)$., Comment: 27 pages

We propose a new measure of deviations from expected utility theory. For any positive number~$e$, we give a characterization of the datasets with a rationalization that is within~$e$ (in beliefs, utility, or perceived prices) of expected utility theory. The number~$e$ can then be used as a measure of how far the data is to expected utility theory. We apply our methodology to data from three large-scale experiments. Many subjects in those experiments are consistent with utility maximization, but not with expected utility maximization. Our measure of distance to expected utility is correlated with subjects' demographic characteristics.

In 2010, Matom\"{a}ki investigated the set of $A>1$ such that the integer part of $ A^{c^k} $ is a prime number for every $k\in \mathbb{N}$, where $c\geq 2$ is any fixed real number. She proved that the set is uncountable, nowhere dense, and has Lebesgue measure $0$. In this article, we show that the set has Hausdorff dimension $1$., Comment: 15 pages

We demonstrate a method of micromotion minimization of a trapped ion in a linear Paul trap based on the precision measurement of the ion trapping position displacement due to a stray electric field in the radial plane by ion fluorescence imaging. The amount of displacement in the radial plane is proportional to the strength of a stray electric field. Therefore, we evaluated the micromotion compensation condition by measuring the ion displacements from the ion equilibrium position using two different radial trap frequencies with various combinations of the compensation voltage. The residual electric field uncertainty of this technique reached a few volts per meter. This compensation technique does not depend on the orientation of the incident cooling laser or the detuning and imaging direction. Therefore, this method is suitable for a planar ion trap, a stylus ion trap, which limits the propagation angle of lasers, or miniaturized ion trap systems for sensing and metrological applications., Comment: 7 pages, 5 figures

For every non-integral $\alpha>1$, the sequence of the integer parts of $n^{\alpha}$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $\alpha$, and let $\mathrm{PS}(\alpha)$ denote the set of all those terms. For all $X\subseteq \mathbb{N}$, we say that an equation $y=ax+b$ is solvable in $X$ if the equation has infinitely many solutions of distinct pairs $(x,y)\in X^2$. Let $a,b\in \mathbb{R}$ with $a\neq 1$ and $0\leq b

A Piatetski-Shapiro sequence with exponent $\alpha$ is a sequence of integer parts of $n^\alpha$ $(n = 1,2,\ldots)$ with a non-integral $\alpha > 0$. We let $\mathrm{PS}(\alpha)$ denote the set of those terms. In this article, we study the set of $\alpha$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(\alpha)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $\alpha > 2$ such that $\mathrm{PS}(\alpha)$ contains infinitely many arithmetic progressions of length $3$., Comment: 21 pages

By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $\alpha\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^\alpha}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^\alpha}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, by using some polynomial $p(x)\in\mathbb{Q}[x]$ of degree at most $d$. In particular, the above sequence is an arithmetic progression when $d=1$. In this paper, we show the asymptotic density of such numbers $n$ as above. When $d=1$, the asymptotic density is equal to $1/(k-1)$. Although the common difference $r$ is arbitrarily fixed in the above result, we also examine the case when $r$ is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields., Comment: 43 pages, 1 figure

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \] converges to an algebraic number is approximated by $(e-1/e)\varphi (k)$. Here $\varphi(k)$ denotes Euler's totient function., Comment: 15 pages

We implement nonparametric revealed-preference tests of subjective expected utility theory and its generalizations. We find that a majority of subjects' choices are consistent with the maximization of some utility function. They respond to price changes in the direction subjective expected utility theory predicts, but not to a degree that makes them consistent with the theory. Maxmin expected utility a dds no explanatory power. The degree of deviations from the theory is uncorrelated with demographic characteristics. Our findings are essentially the same in laboratory data with a student population and in a panel survey with a general sample of the U.S. population.

Let $r_k(N)$ be the largest cardinality of a subset of $\{1,\ldots,N\}$ which does not contain any arithmetic progressions (APs) of length $k$. In this paper, we give new upper and lower bounds for fractal dimensions of a set which does not contain $(k,\epsilon)$-APs in terms of $r_k(N)$, where $N$ depends on $\epsilon$. Here we say that a subset of real numbers does not contain $(k,\epsilon)$-APs if we can not find any APs of length $k$ with gap difference $\Delta$ in the $\epsilon \Delta$-neighborhood of the set. More precisely, we show multi-dimensional cases of this result. As a corollary, we find equivalences between multi-dimensional Szemer\'edi's theorem and bounds for fractal dimensions of a set which does not contain multi-dimensional $(k,\epsilon)$-APs.

This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions., Comment: 4 pages

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for some $\alpha>0$. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. Furthermore, we extend Szemer\'edi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence $\{\lfloor{n^a}\rfloor\}_{n\in A}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and every $A\subset\mathbb{N}$ with positive upper density. Using this corollary, we show that the set $\{ \lfloor{\lfloor{p^{1/b}}\rfloor^a}\rfloor \mid \text{$p$ prime} \}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and $b>1$. We also prove that, for every $a\ge2$, the graph of $\{\lfloor{n^a}\rfloor\}_{n=1}^\infty$ does not contain any arithmetic progressions of length $3$., Comment: 19 pages; revised Section 1 and the proof of Theorem A.4 and added Section 2

In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $\Delta>0$ with direction $e\in S^{d-1}$ inside the $\epsilon \Delta$ neighbourhood of $F$ for all $\epsilon>0$, $k\geq 3$ and $e\in S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions., Comment: 7 pages

In this paper, we define new fractal dimensions of a metric space in order to calculate the roughness of a set on a large scale. These fractal dimensions are called upper zeta dimension and lower zeta dimension. The upper zeta dimension is an extension of the zeta dimension introduced by Doty, Gu, Lutz, Elvira, Mayordomo, and Moser. We show that the upper zeta dimension is always a lower bound for the Assouad dimension. Moreover, we apply the upper zeta dimension to the existence of weak arithmetic patches of a given set. Arithmetic patches are higher dimensionalized arithmetic progressions. As a corollary, we get the affirmative solution to a higher dimensional weak analogue of the Erd\H{o}s-Tur\'an conjecture. Here the one dimensional case is proved by Fraser and Yu. As examples, we prove the existence of weak arithmetic patches of the set of all irreducible elements of $\mathbb{Z}[\alpha]$, and the set of all prime numbers of the form $p(p(n))$, where $\alpha$ is an imaginary quadratic integer and $p(n)$ denotes the $n$-th prime number., Comment: 20 pages

We provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction., Comment: 8 pages