Golomb Rulers Matthias Beck San Francisco State University Tristram Bogart Universidad de los Andes Tu Pham UC Riverside arXiv:1110.6154
All Sorts of Golomb Rulers 0 3 5 0 1 3 0 2 3 8 0 1 4 6 Golomb ruler: sequence of distinct integers with distinct pairwise differences Every Golomb ruler comes with a length t and some m + 1 markings Optimal Golomb rulers have minimal length for a given number of markings Perfect Golomb rulers can measure every integer from 0 to t Golomb Rulers Matthias Beck 2
All Sorts of Golomb Rulers 0 3 5 0 1 3 0 2 3 8 0 1 4 6 Every Golomb ruler comes with a length t and m + 1 markings Optimal Golomb rulers have minimal length for a given number of markings Perfect Golomb rulers can measure every integer from 0 to t Fun Exercise: There are no perfect Golomb rulers of lenth t > 6 Golomb Rulers Matthias Beck 2
All Sorts of Golomb Rulers 0 3 5 0 1 3 0 2 3 8 0 1 4 6 Every Golomb ruler comes with a length t and m + 1 markings Research problem: find optimal Golomb rulers with > 26 markings (see http://www.distributed.net/OGR for computational results). Our goal: count all Golomb rulers for given t and m Golomb Rulers Matthias Beck 2
Motivations & Applications Distortion problems in consecutive radio bands − → place radio signals ◮ so that all distances are distinct (Babcock 1950’s) Error-correcting codes ◮ Additive number theory (Sidon sets) ◮ Dissonant music pieces (see Scott Rickard’s TED talk) ◮ Golomb Rulers Matthias Beck 3
Enumeration of Golomb Rulers Goal Study/compute the number g m ( t ) of Golomb rulers of length t with m + 1 markings 0 x t Example g 2 ( t ) = # { x ∈ Z : 0 < x < t, x � = t − x } Golomb Rulers Matthias Beck 4
Enumeration of Golomb Rulers Goal Study/compute the number g m ( t ) of Golomb rulers of length t with m + 1 markings 0 x t Example g 2 ( t ) = # { x ∈ Z : 0 < x < t, t � = 2 x } Golomb Rulers Matthias Beck 4
Enumeration of Golomb Rulers Goal Study/compute the number g m ( t ) of Golomb rulers of length t with m + 1 markings 0 x t Example 1 g 2 ( t ) = # { x ∈ Z : 0 < x < t, t � = 2 x } � t − 1 if t is odd = t − 2 if t is even . . . a quasipolynomial in t 2 t 2 − 4 t + 10 1 if t ≡ 0 , 2 t 2 − 3 t + 5 1 if t ≡ 1 , 5 , 7 , 11 , Example 2 2 2 t 2 − 4 t + 6 1 g 3 ( t ) = (mod 12) if t ≡ 2 , 10 , 2 t 2 − 3 t + 9 1 if t ≡ 3 , 9 , 2 2 t 2 − 4 t + 8 1 if t ≡ 4 , 6 , 8 Golomb Rulers Matthias Beck 4
Enumeration of Golomb Rulers Goal Study/compute the number g m ( t ) of Golomb rulers of length t with m + 1 markings 0 x t Example 1 g 2 ( t ) = # { x ∈ Z : 0 < x < t, t � = 2 x } � t − 1 if t is odd = t − 2 if t is even . . . a quasipolynomial in t Theorem 1 The Golomb counting function g m ( t ) is a quasipolynomial in t 1 of degree m − 1 with leading coefficient ( m − 1)! Golomb Rulers Matthias Beck 4
Let’s start counting. . . 0 x 1 x 2 t � � x ∈ Z 4 : 0 = x 0 < x 1 < x 2 < x 3 = t g 3 ( t ) := # all x j − x k distinct Golomb Rulers Matthias Beck 5
Let’s start counting. . . z 1 z 2 z 3 0 x 1 x 2 t � � x ∈ Z 4 : 0 = x 0 < x 1 < x 2 < x 3 = t g 3 ( t ) := # all x j − x k distinct � � z 1 + z 2 + z 3 = t z ∈ Z 3 = # > 0 : � j ∈ U z j � = � j ∈ V z j for all dpcs U, V ⊂ [3] Golomb Rulers Matthias Beck 5
Let’s start counting. . . z 1 z 2 z 3 0 x 1 x 2 t � � x ∈ Z 4 : 0 = x 0 < x 1 < x 2 < x 3 = t g 3 ( t ) := # all x j − x k distinct � � z 1 + z 2 + z 3 = t z ∈ Z 3 = # > 0 : � j ∈ U z j � = � j ∈ V z j for all dpcs U, V ⊂ [3] where dpcs is shorthand for “disjoint proper consecutive subset,” and [ m ] := { 1 , 2 , . . . , m } . x 1 � = x 2 ⇐ ⇒ z 2 > 0 x 2 � = t − x 1 ⇐ ⇒ z 1 � = z 3 x 2 � = t − x 2 ⇐ ⇒ z 1 + z 2 � = z 3 Golomb Rulers Matthias Beck 5
Let’s start counting. . . z 1 z 2 z 3 0 x 1 x 2 t � � x ∈ Z 4 : 0 = x 0 < x 1 < x 2 < x 3 = t g 3 ( t ) := # all x j − x k distinct � � z 1 + z 2 + z 3 = t z ∈ Z 3 = # > 0 : � j ∈ U z j � = � j ∈ V z j for all dpcs U, V ⊂ [3] where dpcs is shorthand for “disjoint proper consecutive subset,” and [ m ] := { 1 , 2 , . . . , m } . More generally, � � x ∈ Z m +1 : 0 = x 0 < x 1 < · · · < x m − 1 < x m = t g m ( t ) := # all x j − x k distinct � � z 1 + z 2 + · · · + z m = t z ∈ Z m = # > 0 : � j ∈ U z j � = � j ∈ V z j for all dpcs U, V ⊂ [ m ] Golomb Rulers Matthias Beck 5
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Golomb Rulers Matthias Beck 6
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = L ∆ ( t ) = . . . Golomb Rulers Matthias Beck 6
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 Golomb Rulers Matthias Beck 6
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 � t − 1 � L ∆ ( − t ) = 2 Golomb Rulers Matthias Beck 6
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 � t − 1 � L ∆ ( − t ) = = L ∆ ◦ ( t ) 2 For example, the evaluations L ∆ ( − 1) = L ∆ ( − 2) = 0 point to the fact that neither ∆ nor 2∆ contain any interior lattice points. Golomb Rulers Matthias Beck 6
Enter Geometry Lattice polytope P ⊂ R d – convex hull of finitely points in Z d � t P ∩ Z d � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 � t − 1 � L ∆ ( − t ) = = L ∆ ◦ ( t ) 2 Theorem (Ehrhart 1962) L P ( t ) is a polynomial in t . Theorem (Macdonald 1971) ( − 1) dim P L P ( − t ) = L P ◦ ( t ) Golomb Rulers Matthias Beck 6
Enter Geometry Rational polytope P ⊂ R d – convex hull of finitely points in Q d � t P ∩ Z d � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 � t − 1 � L ∆ ( − t ) = = L ∆ ◦ ( t ) 2 Theorem (Ehrhart 1962) L P ( t ) is a quasipolynomial in t . Theorem (Macdonald 1971) ( − 1) dim P L P ( − t ) = L P ◦ ( t ) Golomb Rulers Matthias Beck 6
Enter Geometry Rational polytope P ⊂ R d – convex hull of finitely points in Q d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = � t +2 = 1 � L ∆ ( t ) = 2 ( t + 1)( t + 2) 2 � t − 1 � L ∆ ( − t ) = = L ∆ ◦ ( t ) 2 For 2 -dimensional lattice polygons, Ehrhart–Macdonald’s theorem follows from Pick’s theorem. Golomb Rulers Matthias Beck 6
Enter Geometry Rational polytope P ⊂ R d – convex hull of finitely points in Q d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = ( x, y ) ∈ Z 2 � � L ∆ ( t ) = # ≥ 0 : x + y ≤ t Golomb Rulers Matthias Beck 7
Enter Geometry Rational polytope P ⊂ R d – convex hull of finitely points in Q d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # t Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = t � ( x, y ) ∈ Z 2 � L ∆ ( t ) = # ≥ 0 : x + y ≤ t t ( x, y, z ) ∈ Z 3 � � = # ≥ 0 : x + y + z = t Golomb Rulers Matthias Beck 7
Enter Geometry Rational polytope P ⊂ R d – convex hull of finitely points in Q d t P ∩ Z d � � For t ∈ Z > 0 let L P ( t ) := # t Example: ∆ = conv { (0 , 0) , (1 , 0) , (0 , 1) } ( x, y ) ∈ R 2 : x, y ≥ 0 , x + y ≤ 1 � � = t � ( x, y ) ∈ Z 2 � L ∆ ( t ) = # ≥ 0 : x + y ≤ t t ( x, y, z ) ∈ Z 3 � � = # ≥ 0 : x + y + z = t Hmmm . . . � � z 1 + z 2 + z 3 = t z ∈ Z 3 g 3 ( t ) = # > 0 : � j ∈ U z j � = � j ∈ V z j for all dpcs U, V ⊂ [3] Golomb Rulers Matthias Beck 7
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