| 1. |
- Dukes, P., et al.
(författare)
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On the possible orders of a basis for a finite cyclic group
- 2010
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Ingår i: Electronic Journal of Combinatorics. - 1077-8926 .- 1097-1440. ; 17:1
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Tidskriftsartikel (refereegranskat)abstract
- We prove a result concerning the possible orders of a basis for the cyclic group Z(n), namely: For each k is an element of N there exists a constant c(k) > 0 such that, for all n is an element of N, if A subset of Z(n) is a basis of order greater than n/k, then the order of A is within c(k) of n/l for some integer l is an element of [1, k]. The proof makes use of various results in additive number theory concerning the growth of sumsets. Additionally, exact results are summarized for the possible basis orders greater than n/4 and less than root n. An equivalent problem in graph theory is discussed, with applications.
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| 2. |
- Hegarty, Peter, 1971
(författare)
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A Cauchy-Davenport type result for arbitrary regular graphs
- 2011
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Ingår i: Integers. - 1867-0660. ; 11:2, s. 227-235
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by the Cauchy-Davenport theorem for sumsets, and its interpretation in terms of Cayley graphs, we prove the following main result: There is a universal constant e > 0 such that, if G is a connected, regular graph on n vertices, then either every pair of vertices can be connected by a path of length at most three, or the number of pairs of such vertices is at least 1+e times the number of edges in G. We discuss a range of further questions to which this result gives rise.
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| 3. |
- Hegarty, Peter, 1971
(författare)
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A variant of the discrete isoperimetric problem
- 2004
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Ingår i: Ars Combin.. ; 73
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Tidskriftsartikel (refereegranskat)abstract
- We consider a variant of what is known as the discrete isoperimetric problem, namely the problem of minimising the size of the boundary of a family of subsets of a finite set. We use the technique of $\lq$shifting' to provide an alternative proof of a result of Hart. This technique was introduced in the early 1980s by Frankl and F\"{u}redi and gave alternative proofs of previously known classical results like the discrete isoperimetric problem itself and the Kruskal-Katona theorem. Hence our purpose is to bring Hart's result into this general framework.
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| 4. |
- Hegarty, Peter, 1971
(författare)
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Answers to Two Questions Posed by Farhi Concerning Additive Bases
- 2009
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Ingår i: Journal of Number Theory. ; 129:12, s. 3052-3058
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Tidskriftsartikel (refereegranskat)abstract
- Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.
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| 5. |
- Hegarty, Peter, 1971
(författare)
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Essentialities in additive bases
- 2009
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 137:5, s. 1657-1661
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Tidskriftsartikel (refereegranskat)abstract
- Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.
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| 6. |
- Hegarty, Peter, 1971
(författare)
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On m-covering families of Beatty sequences with irrational moduli
- 2012
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Ingår i: Journal of Number Theory. - : Elsevier BV. - 0022-314X .- 1096-1658. ; 132:10, s. 2277-2296
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Tidskriftsartikel (refereegranskat)abstract
- We generalise Uspensky's theorem characterising eventual exact (e.e.) covers of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for any m \in \N, by homogeneous sequences with irrational moduli. We also consider inhomogeneous sequences, again with irrational moduli, and obtain a purely arithmetical characterisation of e.e. m-covers. This generalises a result of Graham for m = 1, but when m > 1 the arithmetical description is more complicated. Finally we speculate on how one might make sense of the notion of an exact m-cover when m is not an integer, and present a "fractional version" of Beatty's theorem.
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| 7. |
- Hegarty, Peter, 1971
(författare)
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Permutations avoiding arithmetic patterns
- 2004
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Ingår i: Electron. J. Combin.. ; 11:1
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Tidskriftsartikel (refereegranskat)abstract
- A permutation $\pi$ of an abelian group $G$ (that is, a bijection from $G$ to itself) will be said to avoid arithmetic progressions if there does not exist any triple $(a,b,c)$ of elements of $G$, not all equal, such that $c-b=b-a$ and $\pi(c)-\pi(b)=\pi(b)-\pi(a)$. The basic question is, which abelian groups possess such a permutation ? This and problems of a similar nature will be considered.
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| 8. |
- Hegarty, Peter, 1971
(författare)
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The inverse problem for representation functions for general linear forms
- 2008
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Ingår i: Integers. ; 8:Paper A16
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Tidskriftsartikel (refereegranskat)abstract
- The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X such that there are f(x) solutions (counted appropriately) to L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists. This paper represents the first systematic study of this problem for arbitrary linear forms when X = Z, the setting which in many respects is the most natural one. Having first settled on the "right" way to count representations, we prove that every primitive form has a unique representation basis, i.e.: a set A which represents the function f \equiv 1. We also prove that a partition regular form (i.e.: one for which no non-empty subset of the coefficients sums to zero) represents any function f for which {f^{-1}(0)} has zero asymptotic density. These two results answer questions recently posed by Nathanson. The inverse problem for partition irregular forms seems to be more complicated. The simplest example of such a form is x_1 - x_2, and for this form we provide some partial results. Several remaining open problems are discussed.
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| 9. |
- Hegarty, Peter, 1971
(författare)
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The postage stamp problem and essential subsets in integer bases
- 2010
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Ingår i: David and Gregory Chudnovsky (eds.), Additive Number Theory : Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson. Springer-Verlag, New York.. - 9780387370293 ; , s. 153-171
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that E(h,k) = \Theta_{k} ([h^{k}/\log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) \sim (h-1) {\log k \over \log \log k}.
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| 10. |
- Hegarty, Peter, 1971, et al.
(författare)
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The structure of maximum subsets of {1,...,n} with no solutions to a+b=kc
- 2005
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Ingår i: Electron. J. Combin.. ; 12
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Tidskriftsartikel (refereegranskat)abstract
- If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,...,n\}$ for $k\ge 4$ and $n$ large.
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