A characterization of covering equivalence
A characterization of minimal zero-sequences of index one in finite cyclic groups.
A closer look at some new identities.
A combinatorial identity arising from cobordism theory.
A construction for sets of integers with distinct subset sums.
A contribution to infinite disjoint covering systems
Let the collection of arithmetic sequences be a disjoint covering system of the integers. We prove that if for some primes and integers , then there is a such that . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to . The above conjecture holds for saturated systems with such that the product of its prime factors is at most .
A counterexample to a conjecture of Erdős, Graham and Spencer.
A discrete Fourier kernel and Fraenkel's tiling conjecture
A generalization of an IMO problem.
A generalization of Beatty's theorem.
A generalization of Freiman's 3k-3 Theorem
A generalization of Pascal’s triangle using powers of base numbers
In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.
A multiple set version of the 3k-3 theorem.
A note on a problem of Hilliker and Straus.
A note on sets.
A note on minimal zero-sum sequences over ℤ
A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms and negative terms . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive integer n.
A note on -sets.
A note on the lonely runner conjecture.
A note on the number of -sum-free sets.
A problem of Rankin on sets without geometric progressions
A geometric progression of length k and integer ratio is a set of numbers of the form for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence of positive real numbers with a₁ = 1 such that the set contains no geometric progression of length k and integer ratio. Moreover, is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...