arXiv subjects
Publications and source records attributed to Katalin Gyarmati.
We present a concise, elementary proof that the second smallest modulus of a covering system must be bounded, simplifying the known results through a maximality argument.
Starting with Ramanujan's famous taxicab problem, we can study the solvability of the equations $p^n+q^n=r^n+s^n$ and, more generally, $p_1^{k_1}+\dots+p_m^{k_m}=0$ among polynomials.
The primary objective of this section is to demonstrate that the actual pseudorandom measures of our construction are significantly smaller than the theoretical upper bounds derived from the Weil theorem. Regarding the family of sequences, we note that the construction $E_{f,g,h}$ allows for a large variety of sequences by choosing different triples of polynomials. While the detailed analysis of the cross-correlation measure of such a family is a challenging problem and lies beyond the scope of the present paper, the structure of the construction suggests that sequences generated by different polynomials will remain nearly orthogonal. Indeed, since each sequence is built from distinct Legendre symbol sequences with proven low correlation, their combinations are expected to maintain the same level of independence.
Our work is motivated by the fact that the norms of the Eulerian integers are related to the sums of form $a^2-ab+b^2$, providing a natural generalization for problems concerning products over sums or differences of integers. Let $E$ be the set of Eulerian integers. We define $\omega_{\mathbb N}(x)$ as the number of distinct prime divisors of $x\in\mathbb N$, and $\omega_E(x)$ as the number of distinct Euler prime divisors of $x\in E$. By the Erd\H{o}s--Tur\'an theorem, if $\mc A\subset\mathbb Z^{+}$ and $|\mathcal{A}|=3\cdot{2^{k-1}}$ ($k\in\mathbb{Z}^+$), then $\omega_\mathbb{N}(\prod_{a,b\in\mathcal{A},a\neq{b}}(a+b))\geq{k+1}$. We prove that if $\mathcal{A} \subset E$ is a finite set and $\rho \in E$, then the value of $\omega_E(\prod_{a,b \in \mathcal{A}, a \neq b}(a+\rho b))$ has a lower bound of order $\log|\mathcal{A}|$. Consequently, we provide lower bounds for $\mathcal{A} \subset \mathbb{N}$ for both $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ and $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2-ab+b^2))$. We also give an upper bound for the minimum of $\omega_{\mathbb{N}}(\prod_{a,b \in \mathcal{A}, a \neq b}(a^2+ab+b^2))$ with a computer program, if $|\mathcal{A}|\le 8$ and sets whose largest element is relatively small. Furthermore, using a Diophantine number theoretical lemma of Gy\H{o}ry, S\'ark\"ozy, and Stewart, we give a lower bound of order $\log|\mathcal{A}|$ for $\omega_{\mathbb{N}}(\prod_{a \in \mathcal{A}, b \in \mathcal{B}}(f(a,b)))$ for a specific class of polynomials $f \in \mathbb{Z}[x,y]$ and finite sets $\mathcal{A}, \mathcal{B} \subset \mathbb{Z}$.
We study whether several consecutive prime gaps can all be relatively large at the same time, or is it possible that all are squares or perfect powers, or perhaps none of them are squares? A few related results and problems are also presented.
The aim of this paper is to prove a general version of Plünnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices $i_1, ... i_l.$ Then we prove the existence of a non-empty subset $X$ of $A$ such that we have `good control' over the size of the sumset $X+B_1+... +B_k$. As an application of this result we generalize an inequality of \cite{gymr} concerning the submultiplicativity of cardinalities of sumsets.
For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.