Note on a Conjecture of Wegner

Note on a Conjecture of W egner Dominik Kenn F ebruary 2, 2008 Abstract The optimal pac kings of n unit discs in the plane are kno wn for those n ∈ N , which satisfy certain n umber theoretic conditions. Their geometric realizations are the ext r emal Gr o emer p ackings (or We gner p ackings ). But an extremal Groemer pac k ing of n unit discs do es not exist for all n ∈ N and in this case, the number n is called exc eptional . W e are interested in n umber theoretic characterizati ons of the exceptional num bers. A coun terexample is given to a conjecture of W egner concerning suc h a chara cterization. W e f urther give a characterization of the ex ceptional num bers, whose shap e is closely related to th at of W egner’s conjecture. 1 In tro duction It has been conjectured b y L. F. T´ oth [3], that for any g iven n ∈ N , the optimal packing with resp ect to the conv e x h ull of n discs ha s to be a s ubset with n elements of the hexagona l lattice pac king whose c onv ex hull is ”a s simila r to a regular hexag on as p ossible”. In this c ontext it is clear, tha t a packing is not optimal, if a further disc fits into the con v ex hull of the previous discs without int ersecting the interior o f at least one of the pre v ious discs. Therefore we will restrict o urselves to dense pac kings. Beyond this, it suffices to cons ider only unit discs. W egner [6] solved the a bove problem almost completely . But the packing of n unit discs minimizing the a r ea o f the co nv ex hull of the discs is not known for all n ∈ N . W egner prov ed tha t, in case of existence, the so-called extr emal Gr o emer p ackings are optimal and they are the only optimal packings. F or n with n = 1 + 6  a 2  , the extremal Gro emer pac king o f n unit discs exists and indeed has the sha p e of a regula r hexagon and there is no other packing with the s a me densit y . (This has b een part of T´ oth’s conjecture.) If there exists a n extremal Gro emer packing for so me n with n = 1 + 6  a 2  + ab + c , then it has the shape of a degenerated he x agon. In this case, there can be mo re than o ne extremal Gro emer pac king of n unit discs . Now a num be r n is c alled exc eptional , if there is no extremal Gro emer pac king of n unit discs. Since the exis tence of extremal Groe mer pac kings ca n be form u- lated as a n umber theoretic pr oblem, it is natura l to a s k for a n umber theoretic characterization o f the exceptional n umbers . Mor eov er, the knowledge of such a c haracter ization m ay be useful to describ e the geometric shap e of the optimal packings of n unit discs for exceptiona l n umbers n . W egner [5] gav e a conjecture concerning a num be r theor etic characteriza tion of 1 the exceptiona l num bers. B¨ or ¨ oczky and Ruzsa [1] pr oved another character- ization, but surprisingly , they did not compar e their re s ults to the results of W egner. W e prov e here that W egner’s conjecture is wrong and by mea ns of in vestigating the connection betw een the results o f W egner and B¨ o r¨ oc z ky and Ruzsa, we will ”corr ect” W egner ’s conjecture. 2 Notations and Preliminaries W e re call some necess ary facts and definitions. Each n ∈ N ca n b e wr itten a s n = 1 + 6  a 2  + ab + c where the parameter s a, b, c ∈ Z have to b e chosen maximal in this or der. It then follows tha t 1 ≤ a, 0 ≤ b ≤ 5 and 0 ≤ c < a . Set p 0 ( n ) := 6 ( a − 1) + b + 1 − δ 0 ,b + c where δ denotes the Kroneck e r symbol. A packing of n ∈ N unit discs B 2 in Euclidia n space R 2 is denoted by C n + B 2 , where C n is the set of the cen ters of the discs. A packing C n + B 2 is called a Gr o emer p acking , if the elements of C n form a, not necessarily regular , hexagon in R 2 . Let conv ( M ) de no te the con vex hull of a set M a nd ∂ M its b oundary . F or a Gr o emer packing C n + B 2 , a disc B 2 ⊂ C n + B 2 is called a b oundary disc , if ∂ B 2 ∩ ∂ ( conv ( C n + B 2 )) 6 = ∅ . Now let p ( C n + B 2 ) deno te the num ber of b o undary disc s of C n + B 2 . W egner prov e d p ( C n + B 2 ) ≥ p 0 ( n ) for each Gro emer packing. If equa lity holds, the Gro emer packing is called ex tr emal (or a We gn er p acking ). Combining the results o f [4 ] and [2], o ne gets p 0 ( n ) = l √ 12 n − 3 m − 3 . (1) F or n not to o small, eac h set conv ( C n ) r elated to a Gro emer pac king C n + B 2 has six sides, that mea ns there are six line segments cont aining the centers of the bo undary discs. F or ea ch of those line segmen ts, let p i , 1 ≤ i ≤ 6 , denote the num ber of centers lying o n that line segment. Then one gets a sequence p 1 , . . . , p 6 , called the b oundary s e quenc e o f the packing. The boundar y s e q uence is uniquely deter mined b y the pac king up to cyclic p er m utations. Now p i + p i +1 = p i +3 + p i +4 and a simple consideration shows n = ( p 1 + p 2 − 1)( p 3 + p 4 − 1) −  p 1 2  −  p 4 2  (2) and for e x tremal Gr o emer packings, o ne also has p 0 ( n ) = p 1 + 2 p 2 + 2 p 3 + p 4 − 6 . (3) 2 F or each Gro emer packing C n + B 2 with n = 1 + 6  a 2  + ab + c and b and c not both equal 0 , the num bers p 1 , . . . , p 6 hav e to satisfy the following conditions (see [6], pa ge 6): a − 1 2 ≤ p i ≤ 2 a − c . (4) So an extrema l Gro emer pa cking C n + B 2 exists if a nd only if ther e ar e natural nu mbers p 1 , . . . , p 4 with (2), (3) and (4). Those n ∈ N , for which no ex tremal Gro emer packing exists, are called exc eptional numb ers . In [5], W egner gives the following Conjecture 2.1 (W egne r) A numb er n ∈ N is ex c eptional if and only if the p ar ameters a, b, c satisfy one of the fol lowing c onditions: i ) b = 2 and a − c ≡ − 6 m mo d 9 m +1 for m ∈ N 0 ii ) b = 5 and a − c ≡ 6 · 9 m mo d 9 m +1 for m ∈ N 0 . In [1], B¨ or¨ o czky a nd Ruzsa prove the following Theorem 2.2 (B¨ or¨ oczky and Ruzsa) A numb er n ∈ N is ex c eptional if and only if l √ 12 n − 3 m 2 + 3 − 12 n = (3 k − 1 )9 ℓ for some k , ℓ ∈ N . 3 A coun terexample to the W egner c onjecture Prop ositi on 3.1 The We gner c onje ctu r e is wr ong in gener al. Pro of W e refer to par t i) of c o njecture 2.1. Let b = 2 and choo se m = 2. Then we hav e a − c ≡ − 12 ≡ 717 mo d 9 3 . Now for a = 717 and c = 0 we obtain n = 1 + 6  a 2  + ab + c = 1 5415 51 and condition i) claims that this is an a xceptional num b er. By computer-aide d ca lculation, we find (beside other solutions) p 1 = 702 , p 2 = 7 17 , p 3 = 7 14 and p 4 = 741 . The p i ’s satis fy (2 ), (3) and (4 ), so n = 1 54155 1 is no exce ptio nal num b er.  It can be chec ked by an easy calculation, that there are no k , ℓ ∈ N so that the equation from Theore m 2.2 holds for n = 15 41551 , so this num ber is not exceptional in the sense of B¨ or¨ oczky and Ruzsa. In par ticular, co njecture 2 .1 and theorem 2.2 are not equiv alent. Now we g ive a c haracterization o f the e x ceptional n um ber s similar to the W eg ner conjecture by using the re s ult o f B¨ o r¨ oc zky and Ruzsa. F or the sak e of notation, let P k i = j a i = 0 for k < j ∈ Z . 3 Prop ositi on 3.2 A n umb er n ∈ N i s exc eptional if and only if the p ar ameters a, b, c satisfy one of the fol lowing c onditions: i ) b = 2 and a − c ≡ − 6 m mod 9 ℓ , wher e m = ℓ − 2 X i =0 9 i for ℓ with 9 ℓ (3 k − 1) = 12( a − c ) − 9 , ii ) b = 5 and a − c ≡ 6 · 9 ℓ − 1 mo d 9 ℓ , for ℓ with 9 ℓ (3 k − 1) = 12( a − c ) . Pro of The num bers n = 1 + 6  a 2  are not exceptional, s o let b and c b e not b oth eq ua l to 0. F rom Theor em 2.2 it fo llows w ith (1) tha t n ∈ N is exceptional iff 9 ℓ (3 k − 1 ) = 12( a − c ) + ( b − 2) 2 − 9 (5) for some k , ℓ ∈ N . I) Let n b e an exceptiona l n umber. E quation (5) mo dulo 3 yields b = 2 or b = 5, sinc e 0 ≤ b ≤ 5. i) Let b = 2 . Then 4( a − c ) = 9 ℓ k − 3 (9 ℓ − 1 − 1) , therefore k = 4 z for some z ∈ N . Hence a dir ect computation shows a − c = 9 ℓ z − 6 ℓ − 2 X i =0 9 i . Now let m := P ℓ − 2 i =0 9 i . ii) Let b = 5. Then 12( a − c ) = 9 ℓ (3 k − 1 ) , hence 3 k − 1 ≡ 0 mo d 4 and so k ≡ 3 mo d 4. W e write k = 4 z + 3 for some z ∈ N 0 . That yields a − c = 9 ℓ z + 6 · 9 ℓ − 1 . I I) Let a, b, c s a tisfy condition i) or ii). W e show that they yield exce ptio nal nu mbers. i) Let b = 2 and a − c ≡ − 6 m mo d 9 ℓ . F or ℓ, z ∈ N , ther e is a k ∈ N with − 8 ℓ − 2 X i =0 9 i + 12 · 9 ℓ − 1 z − 1 = 9 ℓ − 1 (3 k − 1 ) . 4 This is ea s ily seen by induction. W e wr ite a − c = − 6 m + 9 ℓ z for some z ∈ N . Then ( p 0 ( n ) + 3) 2 + 3 − 12 n = 9 ( − 8 ℓ − 2 X i =0 9 i + 12 · 9 ℓ − 1 z − 1) = 9 ℓ (3 k − 1 ) . This is equa tion (5) with b = 2. ii) Le t b = 5 and a − c ≡ 6 · 9 ℓ − 1 mo d 9 ℓ . W e write a − c = 6 · 9 ℓ − 1 + 9 ℓ z for some z ∈ N 0 and get ( p 0 ( n ) + 3) 2 + 3 − 12 n = 9 ℓ (3(4 z + 3) − 1) . This is equa tion (5) with b = 5 and k := 4 z + 3.  Ac kno wledgemen t I am gr ateful to Prof. A. Sarti for man y helpfull discussions. The final publication is av ailable a t s pringerlink.co m, DOI 10 .1 007/ s1336 6-011-0004-3. References [1] K. J. B¨ o r¨ o czky and I. Z. Ruzsa. Note on an Inequality of Wegner. Discr ete Comput. Ge om. , 37:2 45–2 49, 2007 . [2] H. Ha r b orth. L¨ o s ung zum Pr oblem 6 6 4 A. Elem. Math. , 29:14 –15, 1 974. [3] L. F. T´ o th. Res e arch Problem 13. Perio d. Math. Hun gar. , 6:1 97–19 9, 1975. [4] G. W egne r . Zur K ombinatorik von Kr eispackungen. Kol l. Diskr. Ge om. , 2:225– 230, 198 0. [5] G. W egner. Extremale Gro emerpa ckungen. Studia Sci. Math. Hu ngar. , 19:299 –302 , 19 84. [6] G. W egner. ¨ Uber endlic he Kreispackungen in der Eb ene. Stu dia Sci. Math. Hungar. , 21 :1 –28, 1 986. Dominik Kenn, Ins titut f ¨ ur Mathematik, J ohannes-Gutenberg -Universit¨ at Mainz, Staudingerweg 9, 55 099 Mainz dominik.kenn@go o glemail.com 5