Dejeans conjecture holds for n >= 30

Dejean’s conjecture hol ds for n ≥ 30 James Curri e ∗ and Narad Ramp ersad † Department of Mathemat ics and Sta tisti cs Universit y of Winnip eg 515 Portag e Aven ue Winnip eg , Manito ba R3B 2E9 (C anada ) j.cu rrie @uwi nnipeg.ca n.ra mper sad@ uwinnipeg.ca No vem ber 13, 201 8 Abstract W e extend Carpi’s results b y sho wing that Dejean’s conjecture holds for n ≥ 30 . 1 In tro duction Rep etitions in words ha v e b een studied starting with Th ue [12, 13] at the b eginning of the previous cen tury . Muc h study has also b een giv en to repe- titions with fractional exponen t [1, 3 , 4 , 5, 6, 8]. If n > 1 is an in teger, then an n -pow er is a non-empt y w ord x n , i.e., word x repeated n times in a row. F or rational r > 1, a fractional r -p o w er is a non- empty w ord w = x ⌊ r ⌋ x ′ suc h that x ′ is the prefix of x of length ( r − ⌊ r ⌋ ) | x | . F or example, 01010 is a 5 / 2-p o w er. A basic problem is that of identify ing the r ep etitive thr esho ld for eac h a lphab et size n > 1: What is the infimum of r suc h tha t an infinite se quence on n letters exists, not con taining an y r -p ow ers? ∗ The author is supp o rted by an NSERC Discov ery Grant. † The author is supp o rted by an NSERC Postdo ctoral F ellowship. 1 W e call this infimum the r ep etitive thr esh old of an n -letter alphab et, denoted b y R T ( n ). Dejean’s conjecture [4] is that RT ( n ) =    7 / 4 , n = 3 7 / 5 , n = 4 n/ ( n − 1) n 6 = 3 , 4 The v alues RT (2), RT (3), RT (4) w ere established by T hu e, Dejean and P ansiot, resp ectiv ely [12, 4, 11]. Moulin-Ollagnier [10] verifie d Dejean’s con- jecture fo r 5 ≤ n ≤ 1 1, while Mohammad- No ori and Currie [9 ] prov ed the conjecture for 12 ≤ n ≤ 14. An exciting new dev elopmen t has recen tly o ccurred with the w ork of Carpi [3 ], who sho w ed that Dejean’s conjecture holds for n ≥ 33. V erification of the conjecture is now only lac king fo r a finite n um b er of v alues. In the presen t pap er, w e sharp en Carpi’s metho ds to sho w that D ejean’s conjecture holds f or n ≥ 30. 2 Preliminaries The follow ing definitions a re from sections 8 and 9 of [3]: Fix n ≥ 30. Let m = ⌊ ( n − 3) / 6 ⌋ . Let A m = { 1 , 2 , . . . , m } . Let k er ψ = { v ∈ A ∗ m |∀ a ∈ A m , 4 divides | v | a } . (In fact, this is not Carpi’s d efinition of k er ψ , but rather the assertion of his Lemma 9.1.) A w ord v ∈ A + m is a ψ - k ernel rep etition if it has p erio d q and a prefix v ′ of length q such tha t v ′ ∈ k er ψ , ( n − 1)( | v | + 1) ≥ nq − 3 . It will b e con v enien t to hav e the following new definition: If v has perio d q and its prefix v ′ of length q is in k er ψ , w e say that q is a k ernel p erio d of v . As Carpi states at the beginning of section 9 of [3]: By the results of the previous sections, a t least in the case n ≥ 30, in order to construct an infinite w ord on n letters a v o iding factors of any expo nen t larger than n/ ( n − 1), it is sufficien t to find an infinite w ord on the alphab et A m a v oiding ψ -ke rnel rep etitions. F or m = 5, Carpi pro duces suc h an infinite w ord, based on a pap er-folding construction. He thus establishes Dejean’s conjecture for n ≥ 33. In the presen t pap er, w e giv e an infinite w ord on the a lpha b et A 4 a v oiding ψ -k ernel rep etitions. W e th us establish Dejean’s conjecture for n ≥ 30. 2 Definition 1. L et f : A ∗ 4 → A ∗ 4 b e define d by f (1) = 121 , f (2) = 123 , f (3) = 141 , f (4) = 142 . L et w b e the fixe d p oint of f . It is useful to note that the frequenc y matrix of f , i.e., [ | f ( i ) | j ] 4 × 4 =     2 1 0 0 1 1 1 0 2 0 0 1 1 1 0 1     has an in v erse modulo 4. Remark 1. L et q b e a non-ne gative inte ger, q ≤ 19 66 . Fix n = 3 2 . R1: Wor d w c on tain s no ψ -kernel r ep etition v with kernel p erio d q . R2: Wor d w c o n tains n o fa ctor v with k ernel p erio d q such that | v | /q ≥ 35 / 34 . Note that 32 31 − 34 31 q = 35 34 when q = 34 2 3 = 385 1 3 , so neither piece of the remark implies the other. Note also that t he conditions of the remark b ecome less stringent f or n = 30 , 31. One a lso v erifies that 35 34 + 9 2(1967) ≤ 32 31 − 34 31 q for q ≥ 1967. T o show that w con tains no ψ -k ernel rep etitions for n = 30, 31, 32, it th us suffices to verify R1 and t o sho w that w ord w contains no f actor v with k ernel p erio d q ≥ 1967 suc h that | v | /q ≥ 35 / 34 + 9 / 2(1967 ) . (1) The remarks R1 and R2 are v erified b y compute r searc h, so we will con- sider the second part o f this attack . Fix q ≥ 1967, and supp ose that v is a factor of w with k ernel p erio d q , and | v | /q ≥ 35 / 34. W ithout lo ss of gener- alit y , supp ose that no extens ion of v has p erio d q . W rite v = sf ( u ) p where s (resp. p ) is a suffix (resp. prefix) o f the imag e of a letter, and | s | ( resp. | p | ) ≤ 2. If | v | ≤ q + 2, then 3 5 / 34 ≤ ( q + 2) /q and 1 / 34 ≤ 2 /q , for cing q ≤ 6 8. This con tradicts R2. W e will therefore assume that | v | ≥ q + 3. Supp ose | s | = 2. Since | v | ≥ q + 3, write v = s 0 z s 0 v ′ , where | s 0 z | = q . Examining f , w e see tha t the letter a s preceding a n y o ccurrence of s 0 in w is 3 uniquely determined if | s | = 2. It follows that a s v is a factor of w with k ernel p erio d q , con tra dicting the maximalit y of v . W e conclud e that | s | ≤ 1. Again considering f , w e see that if t is any factor of w of length 3, and u 1 t , u 2 t are prefixes of w , then | u 1 | ≡ | u 2 | (mo d 3). Since | v | ≥ q + 3, we conclude that 3 divides q . W rite q = 3 q 0 . Since | s | ≤ 1, | p | ≤ 2 and | v | ≥ q + 3, w e see that | f ( u ) | ≥ q . Th us f ( u ) has a prefix of length q = 3 q 0 whic h is in k er ψ . As the frequenc y matrix of f is in v ertible mo dulo 4, the prefix o f u o f length q 0 is in k er ψ . W e see that | v | q ≤ 3 | u | + 3 3 q 0 = | u | q 0 + 1 q 0 . Lemma 2. L e t s b e a non-ne g a tive inte g e r. If factor v of w has kernel p erio d q , wher e q ≤ 1966 (3 s ) , then | v | q < 35 34 + 3 1966 s − 1 X j =0 3 − j . Pro of: If s = 0, this is implied by R2. Supp ose t > 0 and the result holds for s < t . Supp ose that 1966(3 t − 1 ) < q ≤ 1966(3 t ) and there is a factor v of w suc h that v has k ernel p erio d q . Supp ose that | v | /q ≥ 35 / 34. Without loss of generalit y , supp o se that no extens ion of v has p erio d q . W e hav e se en that there is a factor u of w with k ernel perio d q 0 = q / 3, 1966(3 t − 2 ) < q 0 ≤ 1966(3 t − 1 ) suc h that | v | /q ≤ | u | /q 0 + 1 /q 0 < 35 34 + 3 1966 t − 2 X j =0 3 − j ! + 1 q 0 (b y the induction h ypo thesis) < 35 34 + 3 1966 t − 2 X j =0 3 − j + 1 1966(3 t − 2 ) = 35 34 + 3 1966 t − 2 X j =0 3 − j + 3 1966(3 t − 1 ) = 35 34 + 3 1966 t − 1 X j =0 3 − j .  4 Theorem 3. Wor d w c on tains no factor v with kernel p erio d q such that | v | /q ≥ 35 / 34 + 9 / 2(1966 ) . Pro of: Supp ose that factor v of w ha s k ernel p erio d q suc h that (1) holds. By Remark 1, w e ha ve q ≥ 1966. By the previous lemma, for some non-negativ e s , | v | /q < 35 34 + 3 1966 s − 1 X j =0 3 − j < 35 34 + 3 1966 ∞ X j =0 3 − j = 35 34 + 9 2(1966) .  Corollary 4. Deje an ’s c onje ctur e ho lds for n = 30 , 31 , 32 . App endix: Computer searc h Supp ose that some factor v of w has k ernel p erio d q ≤ 1 9 66 and e ither 31( | v | + 1) ≥ 32 q − 3 or | v | / q ≥ 35 / 34 + 9 / 2(1967) . Without loss of generalit y , taking suc h a v as short as p ossible, w e ma y assume that | v | ≤  32(1966) − 3 31 − 1  = 2029 . (W e also ha v e  (1966)  35 34 + 9 2(1967)  = 2029 . ) If | v | > 3 , v is a factor of f ( u ) for some factor u of w with | u | ≤ ( | v | + 4) / 3 . F or a non-negative in teger r , let g ( r ) = ⌊ ( r + 4) / 3 ⌋ . Since g 7 (2029) = 2 < 3, (here the exponen t denotes it era t ed function comp o sition) word v must be a factor of f 7 ( u ) for some factor u of w , | u | = 2. The w ord u 0 = 2314112 1 142 con tains all 8 factors o f w whic h ha ve length 2. T o establish R1 and R2, o ne th us c hec ks that they hold for the single w ord f 7 ( u 0 ) (whic h is of length 24,057). 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