Search

In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem over finite valuation rings R of order q r which generalize recent results given by Hegyvári and Hennecart (2013). More precisely, we prove that, for related pairs of two-variable functions f ( x , y ) and g ( x , y ) , if A and B are two sets in R ∗ with | A | = | B | = q α , then max { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + Δ ( α ) , for some Δ ( α ) > 0 . [ABSTRACT FROM AUTHOR]

In this paper we will give some characterizations for the boundedness of the maximal operators, potential operators and singular integral operators on the vanishing generalized Morrey spaces V L Π p ( ⋅ ) , φ ( Ω ) with variable exponent p ( ⋅ ) and bounded set Ω. Moreover, we also obtain some corresponding results for unbounded set Ω on such these spaces. [ABSTRACT FROM AUTHOR]

Abstract: For in , a subset of is called a Furstenberg set of type or -set if for each direction in the unit circle there is a line segment in the direction of such that the Hausdorff dimension of the set is greater than or equal to . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for , , we construct a set of Hausdorff dimension not greater than . Since in a previous work we showed that is a lower bound for the Hausdorff dimension of any , with the present construction, the value is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions . [Copyright &y& Elsevier]

Abstract: Text: A More Sums Than Differences (or sum-dominant) set is a finite set with . Though it was believed that the percentage of subsets of that are sum-dominant tends to zero, Martin and OʼBryant proved a positive percentage is sum-dominant. We generalize their result to other sums and differences. We prove that a positive percent of the time for all nontrivial choices of , and give explicit constructions. We construct sets exhibiting different behavior as more sums/differences are taken. We prove that for any m, a positive percentage of the time. We find the limiting behavior of for an arbitrary set A as and an upper bound on k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if are all sum-dominant. We prove that for any k a positive percentage of sets is k-generational, and no set is k-generational for all k. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ERSMlrEAijY;list=UUfJicAn0WSCOS0IZWMy7HsA;index=1;feature=plcp. [Copyright &y& Elsevier]