منابع مشابه
New Upper Bound for Sums of Dilates
For λ ∈ Z, let λ · A = {λa : a ∈ A}. Suppose r, h ∈ Z are sufficiently large and comparable to each other. We prove that if |A + A| 6 K|A| and λ1, . . . , λh 6 2r, then |λ1 ·A+ . . .+ λh ·A| 6 K |A|. This improves upon a result of Bukh who shows that |λ1 ·A+ . . .+ λh ·A| 6 K|A|. Our main technique is to combine Bukh’s idea of considering the binary expansion of λi with a result on biclique dec...
متن کاملSums of Dilates
The λ-dilate of a set A is λ · A = {λa : a ∈ A}. We give an asymptotically sharp lower bound on the size of sumsets of the form λ1 ·A+ · · ·+λk ·A for arbitrary integers λ1, . . . , λk and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plünnecke’s inequality.
متن کاملOn Sums of Dilates
Let k be a positive integer and let A ⊂ Z. We let k · A = {ka : a ∈ A} denote the k-dilation of A, and let kA = A+ · · ·+ A (k-times) be the k-fold sumset of A. We observe that A+ k · A ⊂ A+ kA = (k + 1)A and that, in general, A+ k · A is much smaller than (k + 1)A. It is well known that |(k + 1)A| (k + 1)|A| − k, and that equality holds only if A is an arithmetic progression. Indeed, if A is a...
متن کاملA New Tight Upper Bound on the Entropy of Sums
We consider the independent sum of a given random variable with a Gaussian variable and an infinitely divisible one. We find a novel tight upper bound on the entropy of the sum which still holds when the variable possibly has an infinite second moment. The proven bound has several implications on both information theoretic problems and infinitely divisible noise channels’ transmission rates.
متن کاملA New Bound for Kloosterman Sums
We give generating functions for Gauss sums for finite general linear and unitary groups. For the general linear case only our method of proof is new, but we deduce a bound on Kloosterman sums which is sometimes sharper than Deligne’s bound from algebraic geometry.
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2017
ISSN: 1077-8926
DOI: 10.37236/6576