arXiv subjects
Publications and source records attributed to Kota Saito.
In 2010, Matomäki 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 $1$.
For every non-integral $α>1$, the sequence of the integer parts of $n^α$ $(n=1,2,\ldots)$ is called the Piatetski-Shapiro sequence with exponent $α$, and let $\mathrm{PS}(α)$ 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 $0\leq b<a$, and suppose that the equation $y=ax+b$ is solvable in $\mathbb{N}$. We show that for all $1<α<2$ the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$. Further, we investigate the set of $α\in (s,t)$ so that the equation $y=ax+b$ is solvable in $\mathrm{PS}(α)$ where $2< s <t$. Finally, we show that the Hausdorff dimension of the set is coincident with $2/s$.
A Piatetski-Shapiro sequence with exponent $α$ is a sequence of integer parts of $n^α$ $(n = 1,2,\ldots)$ with a non-integral $α> 0$. We let $\mathrm{PS}(α)$ denote the set of those terms. In this article, we study the set of $α$ so that the equation $ax + by = cz$ has infinitely many pairwise distinct solutions $(x,y,z) \in \mathrm{PS}(α)^3$, and give a lower bound for its Hausdorff dimension. As a corollary, we find uncountably many $α> 2$ such that $\mathrm{PS}(α)$ contains infinitely many arithmetic progressions of length $3$.
By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $α\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^α}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^α}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, 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.
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 $(e-1/e)φ(k)$. Here $φ(k)$ denotes Euler's totient function.
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 $(k,ε)$-APs in terms of $r_k(N)$, where $N$ depends on $ε$. Here we say that a subset of real numbers does not contain $(k,ε)$-APs if we can not find any APs of length $k$ with gap difference $Δ$ in the $εΔ$-neighborhood of the set. More precisely, we show multi-dimensional cases of this result. As a corollary, we find equivalences between multi-dimensional Szemerédi's theorem and bounds for fractal dimensions of a set which does not contain multi-dimensional $(k,ε)$-APs.
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 Erdős-Turán 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 $\mathbb{Z}[α]$, and the set of all prime numbers of the form $p(p(n))$, where $α$ is an imaginary quadratic integer and $p(n)$ denotes the $n$-th prime number.
This short paper gives another proof of the infinitude of primes by using upper box dimension, which is one of fractal dimensions.
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 $1/x^α$ for some $α>0$. 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 Szemerédi'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 $3$.
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 $Δ>0$ with direction $e\in S^{d-1}$ inside the $εΔ$ neighbourhood of $F$ for all $ε>0$, $k\geq 3$ 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.
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 $ε>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $Δ>0$ inside the $εΔ$ 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 $ε$. 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.