arXiv subjects
Publications and source records attributed to Christian Elsholtz.
We show that any proper symmetric two dimensional arithmetic progression contained in the interval $[-T,T]$ which avoids non-zero perfect squares has at most $O_\varepsilon(T^{20/27+\varepsilon})$ elements. This improves on a result of Croot, Lyall and Rice. We also discuss lower bounds for this problem and their connections to bounds for the least quadratic non-residue modulo a prime.
A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erdős and Freedman asked whether the maximal dimension of a Hilbert cube in the set $\cal{S}=\{n^2:\;n\in\mathbb{N}\}$ of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if $H(a_{0}; a_{1}, \ldots, a_{d})\subset \cal{S}\cap [0, N]$, then $d\leq 7 \log\log N$ for all sufficiently large values of $N$. Here we prove that there exist at least $\gg N^{1/8}$ Hilbert cubes $H(a_{0}; a_{1}, a_{2}, a_{3})$ with $a_{0}, a_{1}, a_{2}, a_{3}\in [0,N]$ in the set of squares. Moreover, we prove that for each $i, j\in\{0, 1, 2, 3\}$ with $i<j$, the set $$ \left\{\frac{a_{i}}{a_{j}}:\;H(a_{0}; a_{1}, a_{2}, a_{3})\subset S\right\} $$ is dense in the set of positive real numbers (in the Euclidean topology).
The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information modulo $p$, so that we can successfully apply the sieve. 2) The applications to Hilbert cubes are of interest in their own right: We study the maximal dimension of Hilbert cubes in various multiplicatively defined sets. For the squareful numbers in $[1,N]$ we achieve an upper bound of the dimension of $d=O(\log N)$. The same upper bounds follow for multiplicative semigroups of integers defined by a positive proportion of the primes, and the set of integers representable by an irreducible positive definite binary quadratic form. Eventually, making use of the sun flower lemma we give an improvement on the maximal dimension $d$ of subset sums in the set of pure powers in $[1,N]$.
In this note, we discuss two problems of Sárközy (2001). In particular, we prove an optimal result on sumsets of sparse subset of primes.
We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a classical construction of Behrend from 1946 beyond lower-order factors (in particular, it is the first quasipolynomial improvement). In the setting of $\mathbb{F}_p^n$ for a fixed prime $p$ and large $n$, we prove a lower bound of $(cp)^n$ for some absolute constant $c>1/2$ (for $c = 1/2$, such a bound can be obtained via classical constructions from the 1940s, but improving upon this has been a well-known open problem).
We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and therefore sets containing no full line. A~trivial lower bound $r_p(\mathbb{F}_p^n)\geq(p-1)^n$ is achieved by a hypercube of side length $p-1$ and it is known that equality holds for $n\in\{1,2\}$. We will however show that $r_p(\mathbb{F}_p^3)\geq (p-1)^3+p-2\sqrt{p}$, which is the first improvement in the three dimensional case that is increasing in $p$. We will also give the upper bound $r_p(\mathbb{F}_p^{3})\leq p^3-2p^2-(\sqrt{2}-1)p+2$ as well as generalizations for higher dimensions. Finally we present some bounds for individual $p$ and $n$, in particular $r_5(\mathbb{F}_5^{3})\geq 70$ and $r_7(\mathbb{F}_7^{3})\geq 225$ which can be used to give the asymptotic lower bound $4.121^n$ for $r_5(\mathbb{F}_5^{n})$ and $6.082^n$ for $r_7(\mathbb{F}_7^{n})$.
We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for any fixed integer $m\ge 2$ and sufficiently large $n$ (in terms of $m$), there exists a three-term progression free subset $A\subseteq \mathbb{Z}_m^n$ of size $|A|\ge (cm)^n$ for some absolute constant $c>1/2$. Such a bound for $c=1/2$ can be obtained with a classical construction of Salem and Spencer from 1942, and improving upon this value of $1/2$ has been a well-known open problem (our proof gives $c= 0.54$). Our construction relies on finding a subset $S\subset \mathbb{Z}_m^2$ of size at least $(7/24)m^2$ with a certain type of reducibility property. This property allows us to ``lift'' $S$ to a three-term progression free subset of $\mathbb{Z}_m^n$ for large $n$ (even though the original set $S\subset \mathbb{Z}_m^2$ does contain three-term arithmetic progressions).
It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively.
In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus $p$. Moreover, we show that for all primes $p \equiv 5 \bmod 6$ with $p \leq 41$, the new construction leads to an exponentially larger growth of the affine and projective caps in $\mathrm{AG}(n,p)$ and $\mathrm{PG}(n,p)$. For example, when $p=23$, the existence of caps with growth $(8.0875\ldots)^n$ follows from a three-dimensional example of Bose (1947), and the only improvement had been to $(8.0901\ldots)^n$ by Edel (2004), based on a six-dimensional example. We improve this lower bound to $(9-o(1))^n$.
For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $(\mathbb{Z}_{m}^{n},+)$. Let $r_{k}(\mathbb{Z}_{m}^{n})$ denote the maximal size of a subset of $\mathbb{Z}_{m}^{n}$ without arithmetic progressions of length $k$ and let $P^{-}(m)$ denote the least prime factor of $m$. We construct explicit progression-free sets and obtain the following improved lower bounds for $r_{k}(\mathbb{Z}_{m}^{n})$: If $k\geq 5$ is odd and $P^{-}(m)\geq (k+2)/2$, then \[r_k(\mathbb{Z}_m^n) \gg_{m,k} \frac{\bigl\lfloor \frac{k-1}{k+1}m +1\bigr\rfloor^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor/2}}. \] If $k\geq 4$ is even, $P^{-}(m) \geq k$ and $m \equiv -1 \bmod k$, then \[r_{k}(\mathbb{Z}_{m}^{n}) \gg_{m,k} \frac{\bigl\lfloor \frac{k-2}{k}m + 2\bigr\rfloor^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor/2}}.\] Moreover, we give some further improved lower bounds on $r_k(\mathbb{Z}_p^n)$ for primes $p \leq 31$ and progression lengths $4 \leq k \leq 8$.
Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.
Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$, $p$ prime and $b$ square free.
Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log s_n} \ge \frac{φ(|D|)}{2|D|(1+\log φ(|D|))}\gg \frac{1}{\log \log |D|}, \] improving a lower bound of $\frac{1}{|D|}$ of Richards (1982). In the special case of sums of two squares, we improve Richards's bound of $1/4$ to $\frac{195}{449}=0.434\ldots$. We also generalize Richards's result in another direction and establish a lower bound on long gaps between sums of two squares in certain sparse sequences.
We prove new upper bounds on the number of representations of rational numbers $\frac{m}{n}$ as a sum of $4$ unit fractions, giving five different regions, depending on the size of $m$ in terms of $n$. In particular, we improve the most relevant cases, when $m$ is small, and when $m$ is close to $n$. The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the number of solutions. These "approximate parametrizations" were the key point to enable computer programmes to filter through large number of equations and inequalities. Furthermore, this result leads to new upper bounds for the number of representations of rational numbers as sums of more than $4$ unit fractions.
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of Euclid's theorem, as in this case Fermat's last theorem has a proof that does not use the infinitude of primes. Similarly, we discuss implications of Roth's theorem on arithmetic progressions, Hindman's theorem, and infinite Ramsey theory towards Euclid's theorem. As a consequence we see that Euclid's Theorem is a necessary condition for many interesting (seemingly unrelated) results in mathematics.
Mills proved that there exists a real constant $A>1$ such that for all $n\in \mathbb{N}$ the values $\lfloor A^{3^n}\rfloor$ are prime numbers. No explicit value of $A$ is known, but assuming the Riemann hypothesis one can choose $A= 1.3063778838\ldots .$ Here we give a first unconditional variant: $\lfloor A^{10^{10n}}\rfloor$ is prime, where $A=1.00536773279814724017\ldots$ can be computed to millions of digits. Similarly, $\lfloor A^{3^{13n}}\rfloor$ is prime, with $A=3.8249998073439146171615551375\ldots .$
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of $\log f(f(n))$ for a class of multiplicative functions $f$. In particular, this class contains functions counting ideals of given norm in the ring of integers of an arbitrary, fixed quadratic number field. As a consequence, we determine such maximal orders for several multiplicative $f$ arising as a normalized function counting representations by certain binary quadratic forms. Incidentally, for the non-multiplicative function $r_2$ which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula $$ \max_{n\leq x} \log r_2(r_2(n))= \frac{\sqrt{\log x}}{\log \log x} (c/\sqrt{2}+o(1))$$ with some explicitly given constant $c>0$.