Beatty’s Theorem Almost Beatty Partitions Applications Future Work Almost Beatty Partitions and Optimal Scheduling Problems Xiaomin Li Mentor : A. J. Hildebrand University of Illinois at Urbana-Champaign MAA MathFest Conference August 1, 2019 1/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Credits This talk is based on a joint work with: Junxian Li (University of Gottigen)¨ Yun Xie (University of Washington) A. J. Hildebrand (University of Illinois) It originated with a project in Spring 2018 at the Illinois Geometry Lab at the University of Illinois at Urbana-Champaign. Faculty mentors: A. J. Hildebrand and Ken Stolarsky Graduate student: Junxian Li Undergraduate students: Weiru Chen, Matthew Cho, Jared Krandel, Xiaomin Li and Yun Xie 2/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Outline 1 Beatty’s Theorem Beatty Sequences 2 Almost Beatty Partitions Uspensky’s Theorem Almost Beatty Sequences Almost Beatty Partitions: Construction 1 Almost Beatty Partitions: Construction 2 3 Applications Scheduling Problems 4 Future Work 3/24 Remarks When α ≤ 1, then the elements of Bα are distinct and have density α in N. When α > 1, Bα has repeated elements. Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Definition Definition: Beatty Sequence Given α > 0, define the Beatty sequence Bα as n n o B := b c; n = 1; 2;::: ; α α where bxc denotes the floor function. 4/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Definition Definition: Beatty Sequence Given α > 0, define the Beatty sequence Bα as n n o B := b c; n = 1; 2;::: ; α α where bxc denotes the floor function. Remarks When α ≤ 1, then the elements of Bα are distinct and have density α in N. When α > 1, Bα has repeated elements. 4/24 p Examples of Beatty Sequences Bα with α = 1=3; 1; 2; π;. α = p1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 2 0 1 2 2 3 4 4 5 6 7 7 8 9 9 10 11 α = π 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 1=3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 5/24 p Examples of Beatty Sequences Bα with α = 1=3; 1; 2; π;. p α = 2 0 1 2 2 3 4 4 5 6 7 7 8 9 9 10 11 α = π 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 1=3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 α = 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5/24 p Examples of Beatty Sequences Bα with α = 1=3; 1; 2; π;. α = π 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 1=3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 α = p1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 2 0 1 2 2 3 4 4 5 6 7 7 8 9 9 10 11 5/24 p Examples of Beatty Sequences Bα with α = 1=3; 1; 2; π;. Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 1=3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 α = p1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 2 0 1 2 2 3 4 4 5 6 7 7 8 9 9 10 11 α = π 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 1=3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 α = p1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 α = 2 0 1 2 2 3 4 4 5 6 7 7 8 9 9 10 11 α = π 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 p Examples of Beatty Sequences Bα with α = 1=3; 1; 2; π;. 5/24 Rearranging the last two sequences gives: B1/φ 1 3 4 6 8 9 11 12 14 16 B1/φ2 2 5 7 10 13 15 Observations: The Beatty sequences B1/φ and B1/φ2 partition N. 1/φ + 1/φ2 = 1 (i.e. the densities of the last two sequences sum up to 1). Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples II p Let φ be the golden ratio defined by φ = ( 5 + 1)=2. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B1/φ 1 3 4 6 8 9 11 12 14 16 17 19 21 22 24 25 B1/φ2 2 5 7 10 13 15 18 20 23 26 28 31 34 36 39 41 6/24 Observations: The Beatty sequences B1/φ and B1/φ2 partition N. 1/φ + 1/φ2 = 1 (i.e. the densities of the last two sequences sum up to 1). Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples II p Let φ be the golden ratio defined by φ = ( 5 + 1)=2. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B1/φ 1 3 4 6 8 9 11 12 14 16 17 19 21 22 24 25 B1/φ2 2 5 7 10 13 15 18 20 23 26 28 31 34 36 39 41 Rearranging the last two sequences gives: B1/φ 1 3 4 6 8 9 11 12 14 16 B1/φ2 2 5 7 10 13 15 6/24 1/φ + 1/φ2 = 1 (i.e. the densities of the last two sequences sum up to 1). Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples II p Let φ be the golden ratio defined by φ = ( 5 + 1)=2. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B1/φ 1 3 4 6 8 9 11 12 14 16 17 19 21 22 24 25 B1/φ2 2 5 7 10 13 15 18 20 23 26 28 31 34 36 39 41 Rearranging the last two sequences gives: B1/φ 1 3 4 6 8 9 11 12 14 16 B1/φ2 2 5 7 10 13 15 Observations: The Beatty sequences B1/φ and B1/φ2 partition N. 6/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty Sequences: Examples II p Let φ be the golden ratio defined by φ = ( 5 + 1)=2. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B1/φ 1 3 4 6 8 9 11 12 14 16 17 19 21 22 24 25 B1/φ2 2 5 7 10 13 15 18 20 23 26 28 31 34 36 39 41 Rearranging the last two sequences gives: B1/φ 1 3 4 6 8 9 11 12 14 16 B1/φ2 2 5 7 10 13 15 Observations: The Beatty sequences B1/φ and B1/φ2 partition N. 1/φ + 1/φ2 = 1 (i.e. the densities of the last two sequences sum up to 1). 6/24 Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty’s Theorem: Statement The following theorem shows that the previous observation holds whenever the densities α and β are irrational and sum up to 1. Theorem (Beatty’s Theorem) Let α and β be two positive irrational numbers such that α + β = 1. Then Bα and Bβ form a partition of the positive integers. 7/24 n m () j ≤ < j + 1 and j ≤ < j + 1 α β n m () j < < j + 1 and j < < j + 1 (since α; β are irrational) α β () jα < n < (j + 1)α and jβ < m < (j + 1)β () j < n + m < j + 1 (since α + β = 1) This is a contradiction since n; m; j are integers. Beatty’s Theorem Almost Beatty Partitions Applications Future Work Beatty Sequences Beatty’s Theorem: Proof, Part I Let α and β be two positive irrational numbers such that α + β = 1.