Narrow Results

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if and p^ε < |B|, |C| < |A| < p^1-ε, then |A + B| + |A · C| > p^{δ (ε)}|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from to satisfies |F(A,B)| > p^β for some β = β (α) > α whenever and |A| ~ |B|~ p^α, 0 < α < 1. The exponential sum ∑_{x∈ A,y∈B}ε_p(axy+bx²y²), ab ≠ 0 (mod p), may be estimated nontrivially for arbitrary sets satisfying |A|, |B| > p^ρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3].

Finally it is shown that if p^ε < |A| < p^1-ε, then always |A + A|+|A^-1 + A^-1| > p^δ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution (mod p) of certain rational functions on the primes (cf. [12]).

In this paper, we give lower and upper bounds for the minimal size μ_G(r,s) of the sumset (or product set) of two finite subsets of given cardinalities r,s in a group G. Our upper bound holds for solvable groups, our lower bound for arbitrary groups. The results are expressed in terms of variants of the numerical function κ_G(r,s), a generalization of the Hopf–Stiefel function that, as shown in [6], exactly models μ_G(r,s) for G abelian.

Consider the congruence class R_m(a) = {a + im : i ∈ Z} and the infinite arithmetic progression P_m(a) = {a + im : i ∈ N₀}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b) + R_m(c)R_m(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd(a,b,c,d,m) = 1, then R_m(a)R_m(b) + R_m(c)R_m(d) is equal to the congruence class R_m(ab+cd), and that the sum of products set P_m(a)P_m(b)+P_m(c)P_m eventually coincides with the infinite arithmetic progression P_m(ab+cd).

Let s ≥ 1, A_i ⊆ {1,2,…,N} (1 ≤ i ≤ s) and k ≥ 3 be an odd integer. In this paper, one of the main results is: if , then (a) each of A_i - A_j (1 ≤ i, j ≤ s) contains an arithmetic progression of length k with the same common difference; (b) all sets A_i - A_i (1 ≤ i ≤ s) contain a common arithmetic progression of length k. Finally, we pose several problems for further research.

We prove, in particular, that if $A,\mathrm{\Gamma }\subset {𝔽}_p^{\ast }$, $\left|\mathrm{\Gamma }\right|<p^{3/4}$ are two arbitrary multiplicative subgroups satisfying $A-A\subseteq \mathrm{\Gamma }\bigsqcup \{0\}$, then $\left|A\right|\ll |\mathrm{\Gamma }{|}^{1/3+o(1)}$. Also, we obtain that for any $𝜀>0$ and a sufficiently large subgroup $\mathrm{\Gamma }$ with $\left|\mathrm{\Gamma }\right|\ll p^{1/2-𝜀}$ there is no representation $\mathrm{\Gamma }$ as $\mathrm{\Gamma }=A+B$, where $A$ is another subgroup, and $B$ is an arbitrary set, $\left|A\right|,\left|B\right|>1$. Finally, we study the number of collinear triples containing in a set of ${𝔽}_p$ and prove a variant of sum–product estimate.

In a recent paper by Curran and Goldmakher, the cardinality of $h$-fold sumsets $hA$ was given when $A\subset {\mathbb{Z}}^d$ has $d+2$ elements. In this paper we provide a different method for doing this and obtain a more general result.

We also obtain an upper bound for the value of $\left|hA\right|$ when $A\subset {\mathbb{Z}}^d$ is a set of $d+3$ elements with simplicial hull.

Let A be a non-empty finite set of integers. For any integer k, let $A+k\cdot A=\{a_1+k{a}_2:a_1,a_2\in A\}$. One of the important problems in additive combinatorics is to find the optimal lower bound for the cardinality of the sumset $A+k\cdot A$. Cilleruelo, Silva, and Vinuesa conjectured that, if k is a positive integer and A is a finite set of integers with sufficiently large cardinality, then $|A+k\cdot A|\ge (k+1)|A|-\lceil k(k+2)/4\rceil$. For some values of k this conjecture is already confirmed. In this article, under certain restrictions on set A, we confirm this conjecture for all positive integers k except for those of the form $2^{m_1}{p}^{m_2}$ or $2^{m_1}{3}^{m_2}{5}^{m_3}$, where p is an odd prime number and $m_i\ge 1$ for $i=1,2,3$.