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Register a new userThe additive structure of the squares inside rings
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When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the sets underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in finite fields of prime order. Finally, for sufficiently small finite fields we computationally calculate the minimum for all prime orders.
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Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) $le$ n(S(n -- 1) + 1) for all n $ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $le$ 966 conjecturally, whereas all is known so far is 536 $le$ S(6) $le$ 1836.
In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.
Let $A subset mathbb{Z}^d$ be a finite set. It is known that $NA$ has a particular size ($vert NAvert = P_A(N)$ for some $P_A(X) in mathbb{Q}[X]$) and structure (all of the lattice points in a cone other than certain exceptional sets), once $N$ is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary $A$. Such explicit results were only previously known in the special cases when $d=1$, when the convex hull of $A$ is a simplex or when $vert Avert = d+2$, results which we improve.
Let $R$ be a commutative unitary ring. An idempotent in $R$ is an element $ein R$ with $e^2=e$. The ErdH{o}s-Burgess constant associated with the ring $R$ is the smallest positive integer $ell$ (if exists) such that for any given $ell$ elements (not necessarily distinct) of $R$, say $a_1,ldots,a_{ell}in R$, there must exist a nonempty subset $Jsubset {1,2,ldots,ell}$ with $prodlimits_{jin J} a_j$ being an idempotent. In this paper, we prove that except for an infinite commutative ring with a very special form, the ErdH{o}s-Burgess constant of the ring $R$ exists if and only if $R$ is finite.
We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.
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