On the Existence of $U$-Polygons of Class $cgeq 4$ in Planar Point Sets

ON THE EXISTENCE OF U -POL YGONS OF CLASS c ≥ 4 IN PLANAR POINT SETS CHRISTIAN HUCK Abstra t. F or a nite set U of diretions in the Eulidean plane, a on v ex non-degenerate p olygon P is alled a U -p olygon if ev ery line parallel to a diretion of U that meets a v ertex of P also meets another v ertex of P . W e  haraterize the n um b ers of edges of U -p olygons of lass c ≥ 4 with all their v erties in ertain subsets of the plane and deriv e expliit results in the ase of ylotomi mo del sets. 1. Intr odution The (disrete parallel) X -ra y of a nite subset F of the Eulidean plane in diretion u is the orresp onding line sum funtion that giv es the n um b ers of p oin ts of F on ea h line parallel to u . It w as sho wn in [18, Prop osition 4.6℄ that the on v ex subsets of an algebr ai Delone set Λ are determined b y their disrete parallel X -ra ys in the diretions of a set U of at least t w o pair- wise non-parallel Λ -diretions (i.e., diretions parallel to non-zero in terp oin t v etors of Λ ) if and only if there is no U -p olygon with all its v erties in Λ . By [18 , Lemma 4.5℄, there alw a ys exists a U -p olygon with all its v erties in Λ if U is a set of at most three pairwise non-parallel Λ -diretions. This leads to the question whi h U -p olygons exist with all their v erties in Λ for sets U of four or more pairwise non-parallel Λ -diretions. W e refer the reader to [12 , 15 , 16, 17, 18 , 19℄ for more on disrete tomograph y and [11 ℄ for the role of U -p olygons in geometri tomograph y , where the X -ra y of a ompat on v ex set in a diretion giv es the lengths of all  hords of the set in this dire- tion. Dulio and P eri ha v e in tro dued the notion of lass of a U -p olygon and demonstrated that for planar latti es L the n um b ers of edges of U -p olygons of lass c ≥ 4 with all their v erties in L are preisely 8 and 12 ; f. [10, Theo- rem 12℄. As a rst step b ey ond the ase of planar latties, this text pro vides a generalization of this result to planar sets that are non-degenerate in some sense and satisfy a ertain anit y ondition on nite sales (Theorem 3.1). It turns out that, for these sets Λ , the existene of U -p olygons of lass c ≥ 4 with all their v erties in Λ is equiv alen t to the existene of ertain anely r e gular p olygons with all their v erties in Λ , a problem that w as addressed in [19 ℄. The obtained  haraterization of n um b ers of v erties of U -p olygons of lass c ≥ 4 with all their v erties in Λ an b e expressed in terms of a simple inlusion of real eld extensions of Q and partiularly applies to algebr ai Delone sets , th us inluding ylotomi mo del sets , whi h form an imp ortan t lass of planar mathemati al quasirystals ; f. [ 2, 7℄. F or ylotomi mo del This w ork w as supp orted b y EPSR C via Gran t EP/D058465/1. It is a pleasure to thank the CR C 701 at the Univ ersit y of Bielefeld for supp ort during a sta y in June 2007, where part of the man usript w as written. 1 2 CHRISTIAN HUCK sets Λ , the n um b ers of v erties of U -p olygons of lass c ≥ 4 with all their v er- ties in Λ an b e expressed b y a simple divisibilit y ondition (Corollary 4.1). In partiular, the ab o v e result on lattie U -p olygons of lass c ≥ 4 b y Dulio and P eri is on tained as a sp eial ase (Corollary 4.3(a)). 2. Definitions and preliminaries Natural n um b ers are alw a ys assumed to b e p ositiv e and the set of rational primes is denoted b y P . Primes p ∈ P for whi h the n um b er 2 p + 1 is prime as w ell are alled Sophie Germain primes . W e denote b y P SG the set of Sophie Germain primes. The rst few ones are 2 , 3 , 5 , 11 , 23 , 29 , 41 , 53 , 83 , 89 , 113 , 131 , 173 , 179 , 191 , 233 , 239 , . . . ; see en try A005384 of [20℄ for further details. The group of units of a giv en ring R is denoted b y R × . As usual, for a omplex n um b er z ∈ C , | z | denotes the omplex absolute v alue | z | = √ z ¯ z , where ¯ . denotes the omplex onjugation. Oasionally , w e iden tify C with R 2 . The unit irle { x ∈ R 2 | | x | = 1 } in R 2 is denoted b y S 1 . Moreo v er, the elemen ts of S 1 are also alled dir e tions . F or a diretion u ∈ S 1 , the angle b etwe en u and the p ositive r e al axis is understo o d to b e the unique angle θ ∈ [0 , π ) with the prop ert y that a rotation of 1 ∈ C b y θ in oun ter-lo  kwise order is a diretion parallel to u . F or r > 0 and x ∈ R 2 , B r ( x ) denotes the op en ball of radius r ab out x . A subset Λ of the plane is alled uniformly disr ete if there is a radius r > 0 su h that ev ery ball B r ( x ) with x ∈ R 2 on tains at most one p oin t of Λ . F urther, Λ is alled r elatively dense if there is a radius R > 0 su h that ev ery ball B R ( x ) with x ∈ R 2 on tains at least one p oin t of Λ . Λ is alled a Delone set if it is b oth uniformly disrete and relativ ely dense. A diretion u ∈ S 1 is alled a Λ - dir e tion if it is parallel to a non-zero elemen t of the dierene set Λ − Λ of Λ . F urther, a b ounded subset C of Λ is alled a  onvex subset of Λ if its on v ex h ull on tains no new p oin ts of Λ . A non-singular ane tr ansformation of the Eulidean plane is giv en b y z 7→ Az + t , where A ∈ GL(2 , R ) and t ∈ R 2 . F urther, reall that a homothety of the Eulidean plane is giv en b y z 7→ λz + t , where λ ∈ R is p ositiv e and t ∈ R 2 . A  onvex p olygon is the on v ex h ull of a nite set of p oin ts in R 2 . F or a subset S ⊂ R 2 , a p olygon in S is a on v ex p olygon with all v erties in S . A r e gular p olygon is alw a ys assumed to b e planar, non-degenerate and on v ex. An anely r e gular p olygon is a non-singular ane image of a regular p olygon. In partiular, it m ust ha v e at least 3 v erties. Let U ⊂ S 1 b e a nite set of pairwise non- parallel diretions. A non-degenerate on v ex p olygon P is alled a U -p olygon if it has the prop ert y that whenev er v is a v ertex of P and u ∈ U , the line ℓ v u in the plane in diretion u whi h passes through v also meets another v ertex v ′ of P . Then, ev ery diretion of U is parallel to one of the edges of P ; f. [10, Lemma 5(i)℄. F urther, one an easily see that a U -p olygon has 2 m edges, where m ≥ card( U ) . F or example, an anely regular p olygon with an ev en n um b er of v erties is a U -p olygon if and only if ea h diretion of U is parallel to one of its edges. The follo wing notion of lass of a U -p olygon w as in tro dued b y Dulio and P eri; f. [10, Denition 1℄. F or 0 < c ≤ card( U ) , a U -p olygon P is said to b e of lass c with resp et to U if c is the maximal n um b er of onseutiv e edges of P whose diretions b elong to U . ON U -POL YGONS OF CLASS c ≥ 4 IN PLANAR POINT SETS 3 Denition 2.1. F or a subset Λ ⊂ C , w e denote b y K Λ the in termediate eld of C / Q that is giv en b y K Λ := Q  Λ − Λ  ∪  Λ − Λ  . F urther, w e set k Λ := K Λ ∩ R , the maximal real subeld of K Λ . F or n ∈ N , w e alw a ys let ζ n := e 2 π i/n , as a sp ei  hoie for a primitiv e n th ro ot of unit y in C . Denoting b y φ Euler's totien t funtion, one has the follo wing standard result for the n th ylotomi eld Q ( ζ n ) . F at 2.2 (Gauÿ) . [21 , Theorem 2.5℄ [ Q ( ζ n ) : Q ] = φ ( n ) . The eld exten- sion K n / Q is a Galois extension with A b elian Galois gr oup G ( Q ( ζ n ) / Q ) ≃ ( Z /n Z ) × , wher e a ( mo d n )  orr esp onds to the automorphism given by ζ n 7→ ζ a n . It is w ell kno wn that Q ( ζ n + ¯ ζ n ) is the maximal real subeld of Q ( ζ n ) and is of degree φ ( n ) / 2 o v er Q ; see [ 21℄. Throughout this text, w e shall use the notation K n = Q ( ζ n ) , k n = Q ( ζ n + ¯ ζ n ) , O n = Z [ ζ n ] , O n = Z [ ζ n + ¯ ζ n ] . Note that that O n and O n are the rings of in tegers in K n and k n , resp etiv ely; f. [21, Theorem 2.6 and Prop osition 2.16℄. F or n o dd, one has φ (2 n ) = φ ( n ) b y the m ultipliativit y of the arithmeti funtion φ and th us K n = K 2 n ; f. F at 2.2 . Denition 2.3. F or a set Λ ⊂ R 2 , w e dene the follo wing prop erties: (Alg) [ K Λ : Q ] < ∞ . (A ) F or all nite subsets F of K Λ , there is a non-singular ane transformation Ψ of the plane su h that Ψ( F ) ⊂ Λ . (Hom) F or all nite subsets F of K Λ , there is a homothet y h of the plane su h that h ( F ) ⊂ Λ . Moreo v er, w e all Λ de gener ate if and only if K Λ is a subeld of R . Remark 2.4. F or an y non-degenerate Λ ⊂ R 2 , the eld K Λ is a omplex extension of Q . T rivially , prop ert y (Hom) implies prop ert y (A ). If Λ satises prop ert y (Alg), then one has [ k Λ : Q ] < ∞ , meaning that k Λ is a real algebrai n um b er eld. W e need the follo wing result of Darb oux [9℄ on seond mid-p oin t p olygons, where the midp oint p olygon M ( P ) of a on v ex p olygon P is the on v ex p olygon whose v erties are the midp oin ts of the edges of P ; ompare also [13 , Lemma 5℄ or [11 , Lemma 1.2.9℄. F at 2.5. L et P 0 b e a  onvex n -gon in R 2 with  entr oid at the origin. F or e ah k ∈ N , dene P k := sec( π /n ) M ( P k − 1 ) . Then the se quen e ( P 2 k ) ∞ k =0  onver ges in the Hausdor metri to an anely r e gular p olygon. If, in the situation of F at 2.5, P 0 is a U -p olygon of lass c , then, for all k , P 2 k is a U -p olygon of lass c , whene also R := lim k →∞ P 2 k is a U -p olygon of lass c . This pro v es the next 4 CHRISTIAN HUCK Lemma 2.6. L et U ⊂ S 1 b e a nite set of dir e tions and let 0 < c ≤ card( U ) . Then, ther e exists a U -p olygon of lass c if and only if ther e is an anely r e gular U -p olygon of lass c . Let ( t 1 , t 2 , t 3 , t 4 ) b e an ordered tuple of four distint elemen ts of the set R ∪ {∞} . Then, its r oss r atio h t 1 , t 2 , t 3 , t 4 i is dened b y h t 1 , t 2 , t 3 , t 4 i := ( t 3 − t 1 )( t 4 − t 2 ) ( t 3 − t 2 )( t 4 − t 1 ) , with the usual on v en tions if one of the t i equals ∞ , th us h t 1 , t 2 , t 3 , t 4 i ∈ R . The follo wing prop ert y of ross ratios of slop es s z of elemen ts z ∈ R 2 \ { 0 } is standard. F at 2.7. L et z j ∈ R 2 \ { 0 } , j ∈ { 1 , . . . , 4 } , b e four p airwise non-p ar al lel elements of the Eulide an plane and let A ∈ GL(2 , R ) . Then, one has h s z 1 , s z 2 , s z 3 , s z 4 i = h s Az 1 , s Az 2 , s Az 3 , s Az 4 i . Lemma 2.8. [18, F at 4.7℄ F or a set Λ ⊂ R 2 , the r oss r atio of slop es of four p airwise non-p ar al lel Λ -dir e tions is an element of the eld k Λ . 3. The hara teriza tion Theorem 3.1. F or a non-de gener ate subset Λ of the plane with pr op erty (A ) and an even numb er m ≥ 8 , the fol lowing statements ar e e quivalent: (i) Ther e is a U -p olygon of lass c ≥ 4 in Λ with m e dges. (ii) Ther e is an anely r e gular U -p olygon of lass c ≥ 4 with m e dges for a set U of Λ -dir e tions. (iii) k m/ 2 ⊂ k Λ . (iv) Ther e is an anely r e gular p olygon in Λ with lcm( m/ 2 , 2) e dges. If Λ additional ly fulls pr op erty (Alg) , then the ab ove assertions only hold for nitely many values of m . Pr o of. Diretion (i) ⇒ (ii) immediately follo ws from Lemma 2.6 . F or dire- tion (ii) ⇒ (iii), let P b e an anely regular U -p olygon of lass c ≥ 4 with m edges for a set U of Λ -diretions. There is then a non-singular ane trans- formation Ψ of the plane su h that R m = Ψ( P ) is a regular m -gon. Sine P is a U -p olygon of lass c ≥ 4 for a set U of Λ -diretions and sine, b y F at 2.7 , the ross ratio of slop es of diretions of edges is preserv ed b y non- singular ane transformations, there are four onseutiv e edges of R m whose ross ratio q of slop es of their diretions, sa y arranged in order of inreasing angle with the p ositiv e real axis, is an elemen t of k Λ ; f. Lemma 2.8. Ap- plying a suitable rotation, if neessary , w e ma y assume that these diretions are giv en in omplex form b y 1 , ζ m , ζ 2 m and ζ 3 m ; f. F at 2.7 again. Using sin( θ ) = − e − iθ (1 − e 2 iθ ) / 2 i , one easily alulates that q = (tan( 3 π m/ 2 ) − tan( π m/ 2 ))(tan( 2 π m/ 2 ) − tan( 0 π m/ 2 )) (tan( 3 π m/ 2 ) − tan( 0 π m/ 2 ))(tan( 2 π m/ 2 ) − tan( π m/ 2 )) = sin( 2 π m/ 2 ) sin( 2 π m/ 2 ) sin( π m/ 2 ) sin( 3 π m/ 2 ) = (1 − ζ 2 m/ 2 )(1 − ζ 2 m/ 2 ) (1 − ζ m/ 2 )(1 − ζ 3 m/ 2 ) = 2 + ζ m/ 2 + ¯ ζ m/ 2 1 + ζ m/ 2 + ¯ ζ m/ 2 ∈ k Λ . ON U -POL YGONS OF CLASS c ≥ 4 IN PLANAR POINT SETS 5 This implies that q q − 1 − 2 = ζ m/ 2 + ¯ ζ m/ 2 ∈ k Λ , the latter b eing equiv alen t to (iii). Diretion (iii) ⇒ (iv) is an imme- diate onsequene of [19, Theorem 3.3℄ in onjuntion with the iden tit y k m/ 2 = k lcm( m/ 2 , 2) . F or diretion (iv) ⇒ (i), assume rst that m/ 2 is o dd. Here, w e are done sine ev ery anely regular p olygon in Λ with lcm( m/ 2 , 2) = m edges is a U -p olygon of lass c = m/ 2 with resp et to an y set U of diretions parallel to m/ 2 onseutiv e of its edges. If m/ 2 is ev en, there is an anely regular p olygon P in Λ with lcm( m/ 2 , 2) = m/ 2 edges. A tta h m/ 2 translates of P edge-to-edge to P in the ob vious w a y and onsider the on v ex h ull P ′ of the resulting p oin t set. Clearly , P ′ is a U ′ -p olygon in K Λ of lass c = c ard( U ′ ) with m edges, where U ′ onsists of the m/ 2 pairwise non-parallel Λ -diretions giv en b y the edges and diagonals of P . By prop ert y (A ), there is a non-singular ane transformation Ψ of the plane su h that Ψ( P ′ ) is a p olygon in Λ . Then, Ψ( P ′ ) is a U -p olygon of lass c = card( U ) in Λ with m edges, where U is a set of m/ 2 pairwise non- parallel Λ -diretions parallel to the elemen ts of Ψ( U ′ ) . Assertion (i) follo ws. If Λ additionally has prop ert y (Alg), then k Λ is an algebrai n um b er eld b y Remark 2.4. Th us, the eld extension k Λ / Q has only nitely man y in ter- mediate elds and the assertion follo ws from ondition (iii) in onjuntion with [19, Corollary 2.7, Remark 2.8 and Lemma 2.9℄.  Corollary 3.2. L et L b e a  omplex algebr ai numb er eld with L = L and let O L b e the ring of inte gers in L . L et Λ b e a tr anslate of L or a tr anslate of O L . F urther, let m ≥ 8 b e an even numb er. Denoting the maximal r e al subeld of L by l , the fol lowing statements ar e e quivalent: (i) Ther e is a U -p olygon of lass c ≥ 4 in Λ with m e dges. (ii) Ther e is an anely r e gular U -p olygon of lass c ≥ 4 with m e dges for a set U of Λ -dir e tions. (iii) k m/ 2 ⊂ l . (iv) Ther e is an anely r e gular p olygon in Λ with lcm( m/ 2 , 2) e dges. A dditional ly, the ab ove assertions only hold for nitely many values of m . Pr o of. This follo ws immediately from Theorem 3.1 in onjuntion with the fat that Λ has prop erties (A ) and (Alg) with K Λ = L ; f. [19 , Setion 3℄.  Remark 3.3. In partiular, Corollary 3.2 applies to translates of omplex ylotomi elds and their rings of in tegers, resp etiv ely , with l = k n for a suitable n ≥ 3 ; f. F at 2.2 and also ompare the equiv alenes of Corollary 4.1 b elo w. 4. Applia tion to ylotomi model sets Delone subsets of the plane satisfying prop erties (Alg) and (Hom) w ere in- tro dued as algebr ai Delone sets in [18 , Denition 4.1℄. Note that algebrai Delone sets are alw a ys non-degenerate, sine this is true for all relativ ely dense subsets of the plane. Examples of algebrai Delone sets are the so- alled ylotomi mo del sets Λ ; f. [18 , Prop osition 4.31℄. By denition, an y ylotomi mo del set Λ is on tained in a translate of O n , where n ≥ 3 , in 6 CHRISTIAN HUCK whi h ase the Z -mo dule O n is alled the underlying Z -mo dule of Λ . More preisely , for n ≥ 3 , let . ⋆ : O n → ( R 2 ) φ ( n ) / 2 − 1 b e an y map of the form z 7→ ( σ 2 ( z ) , . . . , σ φ ( n ) / 2 ( z )) , where the set { σ 2 , . . . , σ φ ( n ) / 2 } arises from G ( K n / Q ) \ { id , ¯ . } b y  ho os- ing exatly one automorphism from ea h pair of omplex onjugate ones; f. F at 2.2 . Then, for an y su h  hoie, ea h translate of the set { z ∈ O n | z ⋆ ∈ W } , where W ⊂ ( R 2 ) φ ( n ) / 2 − 1 is a suien tly `nie' set with non- empt y in terior and ompat losure, is a ylotomi mo del set with under- lying Z -mo dule O n ; f. [16, 17 , 18, 19℄ for more details and prop erties of (ylotomi) mo del sets. Sine O n = O 2 n for o dd n , w e migh t restrit our- selv es to v alues n 6≡ 2 (mo d 4) when dealing with ylotomi mo del sets with underlying Z -mo dule O n . With the exeption of the rystallographi ases of translates of the square lattie O 4 and translates of the triangu- lar lattie O 3 , ylotomi mo del sets are ap erio di (they ha v e no non-zero translational symmetries) and ha v e long-range order; f. [18, Remark 4.23℄. W ell-kno wn examples of ylotomi mo del sets with underlying Z -mo dule O n are the v ertex sets of ap erio di tilings of the plane lik e the Ammann- Beenk er tiling [1, 4 , 14℄ ( n = 8 ), the Tübingen triangle tiling [ 5 , 6℄ ( n = 5 ) and the shield tiling [14 ℄ ( n = 12 ); f. Figure 1 for an illustration. F or de- nitions of the ab o v e v ertex sets of ap erio di tilings of the plane in algebrai terms, w e refer the reader to [17, Setion 1.2.3.2℄ or [16 ℄. As an immediate onsequene of Theorem 3.1 in onjuntion with [19, Corollary 4.1℄ and the iden tit y k m/ 2 = k lcm( m/ 2 , 2) , one obtains the follo wing Corollary 4.1. L et m, n ∈ N with m ≥ 8 an even numb er and n ≥ 3 . F urther, let Λ b e a ylotomi mo del set with underlying Z -mo dule O n . The fol lowing statements ar e e quivalent: (i) Ther e is a U -p olygon of lass c ≥ 4 in Λ with m e dges. (ii) Ther e is an anely r e gular U -p olygon of lass c ≥ 4 with m e dges for a set U of Λ -dir e tions. (iii) k m/ 2 ⊂ k Λ . (iv) Ther e is an anely r e gular p olygon in Λ with lcm( m/ 2 , 2) e dges. (v) k m/ 2 ⊂ k n . (vi) m ∈ { 8 , 12 } , or K m/ 2 ⊂ K n . (vii) m ∈ { 8 , 12 } , or m | 2 n , or m = 4 d with d an o dd divisor of n . (viii) m ∈ { 8 , 12 } , or O m/ 2 ⊂ O n . (ix) O m/ 2 ⊂ O n . Remark 4.2. Com bining Corollary 4.1 and F at 2.7 , one sees that the ross ratios of slop es of diretions of edges of U -p olygons of lass c ≥ 4 in y- lotomi mo del sets Λ , sa y arranged in order of inreasing angle with the p ositiv e real axis, easily follo w from a diret omputation with a nite n um- b er of regular p olygons; f. [12 , 8℄ for deep insigh ts in to this in the ase of planar latties. The follo wing onsequene follo ws immediately from Corollary 4.1 in on- juntion with [19, Corollary 4.2℄. Restrited to v alues n 6≡ 2 (mo d 4) , it deals with the t w o ases where the degree φ ( n ) / 2 of k n o v er Q is either 1 or a prime n um b er p ∈ P ; f. [19, Lemma 2.10℄. ON U -POL YGONS OF CLASS c ≥ 4 IN PLANAR POINT SETS 7 Figure 1. An U -iosagon of lass c = card( U ) = 10 in the v ertex set Λ TTT of the Tübingen triangle tiling with resp et to the set U of Λ TTT -diretions giv en b y the edges and diag- onals of the en tral regular deagon. Corollary 4.3. L et m, n ∈ N with m ≥ 8 an even numb er and n ≥ 3 . F urther, let Λ b e a ylotomi mo del set with underlying Z -mo dule O n . Then, one has: (a) If n ∈ { 3 , 4 } , ther e is a U -p olygon of lass c ≥ 4 in Λ with m e dges if and only if m ∈ { 8 , 12 } . (b) If n ∈ { 8 , 9 , 12 } ∪ { 2 p + 1 | p ∈ P SG } , ther e is a U -p olygon of lass c ≥ 4 in Λ with m e dges if and only if  m ∈ { 8 , 12 , 2 n } , if n = 8 or n = 12 , m ∈ { 8 , 12 , 2 n, 4 n } , otherwise. Example 4.4. As men tioned ab o v e, the v ertex set Λ TTT of the Tübingen triangle tiling is a ylotomi mo del set with underlying Z -mo dule O 5 . By Corollary 4.3 there is a U -p olygon of lass c ≥ 4 in Λ TTT with m edges if and only if m ∈ { 8 , 10 , 12 , 20 } ; see Figure 1 for an U -iosagon of lass c = 10 in Λ TTT . A kno wledgements It is a pleasure to thank Mi hael Baak e, Ri hard J. Gardner and Uw e Grimm for their o op eration and for useful hin ts on the man usript. Referenes [1℄ R. Ammann, B. 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W ashington, Intr o dution to Cylotomi Fields , 2nd ed., Springer, New Y ork, 1997. Dep ar tment of Ma thema tis and St a tistis, The Open University, W al ton Hall, Mil ton Keynes, MK7 6AA, United Kingdom E-mail addr ess : .hukopen.a.uk