We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if variables $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the filed of residues. Also the paper contains a proof of the result that states that if a subgroup $G$ can be presented as a set of values of the polynomial $P(x,y)$, where $x\in A$, and $y\in B$ then the cardinalities of sets $A$ and $B$ are close (in order) to a square root of the cardinality of subgroup $G$.
The paper is devoted to some applications of Stepanov method. In the first part of the paper we obtain the estimate of the cardinality of the set, which is obtained as an intersection of additive shifts of some different subgroups of F^*_p. In the second part we prove a new upper bound for Heilbronn's exponential sum and obtain a series of applications of our result to distribution of Fermat quotients. Also we study additive decompositions of multiplicative subgroups.
In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that in symmetric case G is always an additive basis of order five, provided by |G| > p^{1/2} log^{1/3} p. Also the method allows us to give a new upper bound for Heilbronn's exponential sum.
Let $A$ be a multiplicative subgroup of $\mathbb Z_p^*$. Define the $k$-fold sumset of $A$ to be $kA=\{x_1+\dots+x_k:x_i \in A,1\leq i\leq k\}$. We show that $6A\supseteq \mathbb Z_p^*$ for $|A| > p^{\frac {11}{23} +ε}$. In addition, we extend a result of Shkredov to show that $|2A|\gg |A|^{\frac 85-ε}$ for $|A|\ll p^{\frac 59}$.
Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $Γ_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in H\setminus\{0\}$. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the \emph{prime sum graphs}, in the case where $H=pG$ for some prime number $p$. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph $Γ^+_{G,H}$, which we refer to as the \emph{…
For set $A\subset {\mathbb {F}_p}^*$ define by ${\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ We prove that if ${\mathsf{sf}} (A)-|A|/3$ is small then the set $A$ must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in ${\mathbb {F}_p}$.
We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and prove a "dual" sum-products estimate.
A perfect code $C$ in a graph $Γ$ is an independent set of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $Γ$ is a set of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of…
We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalisations of results of C. Dartyge and A. Sarkozy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such the Karatsuba bound of double character sums and some results from additive combinatorics.
A subset $C$ of the vertex set of a graph $Γ$ is called a perfect code of $Γ$ if every vertex of $Γ$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditi…
Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $σ_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the corresponding zeta-function on groups of $p$-rank bounded by some constant $r$ is rational with a simple denominator. We also provide two explicit examples in rank $r=3$ and $r=4$ as well as a closed formula for $σ_a(F)$.
We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality and the new best known additive energy bound $\mathsf E(A)\lesssim_M |A|^{49/20}$, which aligns, in a sense to be discussed…
Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Maróti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the …
Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then either A+B=G, or the complement of A+B in G is contained in a coset of a subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate conditions for the containment to be strict, and establish a refinement in the case where the sizes of A and B differ significantly.
We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromov's proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be n…
We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem.
