New bounds for (weak) sequenceability in $\mathbb{Z}_k$

NEW BOUNDS F OR (WEAK) SEQUENCEABILITY IN Z k SIMONE COST A AND STEF ANO DELLA FIORE Abstract. A famous conjecture of Graham asserts that ev ery set A ⊆ Z p \ { 0 } can b e ordered so that all partial sums are distinct. Although this conjecture was recently prov ed for sufficien tly large primes by Pham and Sauermann in [16], it remains op en for general ab elian groups, even in the cyclic case Z k . F or cyclic groups, the best kno wn result is due to Bedert and Kravitz in [4], who pro v ed - using a rectification and a t wo-step probabilistic approac h - that the conjecture holds for an y subset A ⊆ Z k \ { 0 } suc h that | A | ≤ exp  c (log p ) 1 / 4  , for some constant c > 0, where p denotes the least prime divisor of k . In this pap er, w e improv e their b ound using a rectification argument again, follow ed by a one-shot probabilistic approac h, showing that the conjecture holds whenever | A | ≤ exp  c (log p ) 1 / 3  , th us improving the exp onent 1 / 4 from [4]. Moreo ver, the same one-shot approach adapts to the t -weak setting: by imp osing all lo cal con- strain ts at once and applying the Lov´ asz Lo cal Lemma, we obtain the existence of a t -weak se- quencing whenev er t ≤ exp  c (log p ) 1 / 4  . 1. Introduction Let A b e a finite subset of an ab elian group ( G, +). W e say that an ordering a 1 , . . . , a | A | of A is valid if its partial sums p 1 = a 1 , p 2 = a 1 + a 2 , . . . , p | A | = a 1 + · · · + a | A | are pairwise distinct. Moreo v er, this ordering is a se quencing if it is v alid and p i  = 0 for ev ery 1 ≤ i ≤ | A | − 1. In this case, w e sa y that A is sequenceable. If w e relax the definition, w e say that an ordering is a t -we ak se quencing if for ev ery i  = j with 1 ≤ | i − j | ≤ t , the partial sums p i and p j are non-zero and distinct. In this case w e sa y that A is t -w eak sequenceable. In the literature, there are several conjectures ab out v alid orderings and sequenceability . W e refer to [9, 13, 15] for an ov erview of the topic, [1 – 3, 8] for lists of related conjectures, and [5] for a treatmen t using rain b ow paths. Here, w e explicitly recall Graham’s conjecture, whic h states that every set of nonzero elements of Z p has a v alid ordering. Conjecture 1.1 ([11] and [10]) . L et p b e a prime. Then every subset A ⊆ Z p \ { 0 } has a valid or dering. Un til recen tly , the main results on this conjecture were for small v alues of | A | ; in particular, in [8], the conjecture was prov ed for sets A of size at most 12. The first result inv olving arbitrarily large sets A w as presented by Kravitz [12], who used a rectification argument to sho w that Graham’s conjecture holds for all sets A of size | A | ≤ log p/ log log p . A similar argumen t was also prop osed (but not published) b y Will Sa win [17]. Then, in [4], Bedert and Kravitz improv ed - using a rectification and a tw o-step probabilistic approac h - this upp er b ound to the follo wing: 2020 Mathematics Subje ct Classific ation. 11B75. Key words and phr ases. Sequenceability , Rectification, Lov´ asz Local Lemma. 1 2 NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k Theorem 1.2 ([4]) . L et p b e a lar ge enough prime and let c > 0 . Then every subset A ⊆ Z p \ { 0 } is se quenc e able pr ovide d that | A | ≤ exp  c (log p ) 1 / 4  . They explicitly state their result for Z p , but their approach can b e easily adapted to a generic cyclic group. Finally , in [16], Graham’s conjecture w as pro ved for all sufficiently large primes p . This result is a consequence of anticoncen tration inequalities developed using a discrete F ourier approach that seems hard to adapt to the cyclic case. In this pap er, w e record t w o complementary developmen ts. In one direction, we show how it is p ossible to improv e the b ound [4] again using a rectification argument and a one-shot probabilistic approac h, and obtain Theorem 1.3 (Improv ed classical b ound) . Ther e exists a c onstant c > 0 such that, denote d by p the le ast prime divisor of k , then every subset A ⊆ Z k \ { 0 } is se quenc e able pr ovide d that | A | ≤ exp  c (log p ) 1 / 3  . In particular, the one-shot sc heme remov es the need to separate the treatmen t of Type I and T ype I I in terv als, and this structural simplification is what ultimately p ermits the sharp er quantitativ e b ound. Here, w e also sho w ho w this approac h can b e lo calized for the t -weak sequenceability problem. F or this problem, in [8], the authors prov ed that if the order of a group is pe then all sufficiently large subsets of the non-identit y elements are t -w eakly sequenceable when p > 3 is prime, e ≤ 3 and t ≤ 6. Then in [6], using a hybrid approac h that com bines Ramsey theory and the probabilistic metho d, the authors pro v ed that if the size of a s ubset A of an ab elian group G is at least t αt for some α > 2 and A do es not con tain 0, then A is t -weak sequenceable. Here, using a one-shot pro cedure that in v olv es the Lov´ asz Lo cal Lemma, w e obtain: Theorem 1.4. Ther e exists a c onstant c > 0 such that, if p is the le ast prime divisor of k , then every subset A ⊆ Z k \ { 0 } is t -we ak se quenc e able pr ovide d that t ≤ exp  c (log p ) 1 / 4  . Notation. F or a sequence b = b 1 , . . . , b r , let IS( b ) := { b 1 + · · · + b j : 0 ≤ j ≤ r } denote the set of initial segment sums of b , and let b := b r , . . . , b 1 denote the reverse of b . In addition, w e denote IS t ( b ) := { b 1 + · · · + b j : 0 ≤ j ≤ t } . Using standard asymptotic notation, w e say that f = Θ( g ) if there exist t w o absolute constan ts C 1 , C 2 > 0 suc h that C 1 g ≤ f ≤ C 2 g . W e also define f ( p ) = o ( g ( p )) if lim p →∞ f ( p ) /g ( p ) = 0. 2. Proof of Theorem 1.4 and the one-shot framework Let G b e an ab elian group. A subset D = { d 1 , . . . , d r } ⊆ G is disso ciate d if ϵ 1 d 1 + · · · + ϵ r d r  = 0 for all ( ϵ 1 , . . . , ϵ r ) ∈ {− 1 , 0 , 1 } r \ { (0 , . . . , 0) } . Equiv alently , D is disso ciated if all of the 2 | D | subset sums of D are distinct. The dimension of a subset B ⊆ G , written dim( B ), is the size of the largest disso ciated set con tained in B . The span( B ) of a subset B ⊆ G , is defined as span( B ) := ( X b ∈ B ϵ b b : ϵ b ∈ {− 1 , 0 , 1 } ) NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k 3 2.1. Structure Theorem. W e b egin by stating a v ariation of the Structure Theorem of Bedert and Kravitz, enunciated here for the rings Z k in a form that also enforces the disso ciated sets D j to ha v e comparable size. Throughout the t -w eak part of this pap er w e set R := R ( k ) = c 1 (log p ) 1 / 2 , where c 1 > 0 is a sufficiently small absolute constant and p is the least prime divisors of k . The Structure Theorem is based on the following rectification Lemma, prov ed here for cyclic groups. Lemma 2.1. If B ⊆ Z k is a nonempty subset of dimension dim ( B ) < R , then ther e is some λ ∈ Z × k such that the dilate λ · B is c ontaine d in the interval ( − k 100 | B | , k 100 | B | ) . Pr o of. Let D b e a maximal disso ciated set of B and let Λ = { 0 , 1 , . . . , p − 1 } ⊂ Z k . It is clear that, giv en λ 1 , λ 2 ∈ Λ, λ 1 − λ 2 ∈ Z × k . Due to the pigeonhole principle, there exists distinct λ 1 , λ 2 ∈ Λ such that ∥ λ 1 x i − λ 2 x i ∥ ≤ k p 1 /r for all i ∈ [1 , r ]. Set λ : = λ 1 − λ 2 , we hav e that λ ∈ Z × k and λ ( D ) ⊆ [ − k p 1 / dim( B ) , k p 1 / dim( B ) ]. Since B ⊆ span( D ), we hav e B ⊆  − dim( B ) k p 1 / dim( B ) , dim( B ) k p 1 / dim( B )  . The thesis follo ws if w e pro v e that (1) dim( B ) k p 1 / dim( B ) < k 100 | B | as long as c 1 is c hosen to b e sufficien tly small. Set h = dim ( B ), Equation (1) is equiv alen t to log( h k p 1 /h ) = log( h ) + log k − 1 /h log p < log k − l og 100 − log | B | . Since | B | < 3 dim( B ) , this relation is implied by h log h + 2 h log 10 + h 2 log 3 < log p. Whic h holds since w e ha v e assumed that h < R = c 1 (log p ) 1 / 2 . □ Theorem 2.2 (Structure Theorem [4]) . F or every nonempty subset A ⊆ Z k \ { 0 } , ther e is some λ ∈ Z × k such that λ · A c an b e p artitione d as λ · A = P ∪ N ∪  s [ j =1 D j  , wher e: (i) the “p ositive” set P is c ontaine d in  0 , k 4 | P ∪ N |  , the “ne gative” set N is c ontaine d in  − k 4 | P ∪ N | , 0  , and the element δ := P s j =1 P d ∈ D j d is c ontaine d in ( − k 4 , k 4 ) ; and if s > 0 then: (ii) P ∪ N is nonempty, and e ach D j is disso ciate d of size | D j | = Θ( R ) ; (iii) δ / ∈ { 0 } ∪ − P ∪ − N , and mor e over δ  = − P p ∈ P p if N is nonempty and δ  = − P n ∈ N n if P is nonempty; (iv) D 1 ∪ D s ∪ { δ } is disso ciate d. 4 NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k 2.2. Ordering P and N . Here we recall the following imp ortant prop osition from [4]. Prop osition 2.3. L et P ⊆ (0 , k 4 | P ∪ N | ) and N ⊆ ( − k 4 | P ∪ N | , 0) b e subsets of Z k , and let δ > 0 b e c on- taine d in (0 , k 4 | P ∪ N | ) ; mor e over, assume that δ  = − P n ∈ N n if P  = ∅ . L et Y + 1 , . . . , Y + m , Y − 1 , . . . , Y − m ⊆ Z b e finite sets. Then ther e ar e or derings p of P and n of N such that p , δ, n is a se quencing and we have | IS( p ) ∩ Y + j | ≤ inf L ∈ N | Y + j | L + L + 4 + 4 j − 1 X i =1 | Y + i | ! and | IS( n ) ∩ Y − j | ≤ inf L ∈ N | Y − j | L + L + 4 + 4 j − 1 X i =1 | Y − i | ! for al l 1 ≤ j ≤ m . 2.3. Splitting, rearrangemen t, a nd a one-shot control of T yp e I and Type I I. W e b egin with the standard an ti-concen tration input for disso ciated sets. Lemma 2.4 (Lemma 5.1 of [4]) . L et D ⊂ G b e a disso ciate d set, and let D = D (1) ⊔ D (2) ⊔ D (3) ⊔ D (4) b e a uniformly r andom p artition of D into four sets of e qual size. Then for every pr op er subset I ⊂ [4] and every x ∈ G , P  X i ∈ I X d ∈ D ( i ) d = x  ≤  | D | | D | · | I | / 4  − 1 ≤  | D | | D | / 4  − 1 . Starting from a set A , we consider D 1 , . . . , D s to b e the disso ciated sets app earing in the struc- tural decomp osition of λ · A provided by Theorem 2.2 and P ⊆ (0 , k 4 | P ∪ N | ), N ⊆ ( − k 4 | P ∪ N | , 0) the sets of p ositive and negative elements there defined. W e split and rearrange the disso ciated sets as follo ws. (S1) F or eac h j ∈ { 2 , . . . , s − 1 } , we partition D j in to four equal parts D j = D (1) j ⊔ D (2) j ⊔ D (3) j ⊔ D (4) j uniformly at random as in Lemma 2.4. W e do all these splittings indep enden tly . (S2) F or the endp oint blo c ks D 1 and D s , we c ho ose a uniform r andom p ermutation σ 1 of D 1 and define D (1) 1 , . . . , D (4) 1 as the four consecutive segments (equal size) of the list σ 1 . Lik ewise, w e choose a uniform random p ermutation σ s of D s and define D (1) s , . . . , D (4) s as consecutive segmen ts. Next, w e place these newly formed sets, together with P and N , in the deterministic order P , D (1) 1 , D (2) 1 , D (1) 2 , D (2) 2 , . . . , D (1) s , D (2) s , D (3) 1 , D (4) 1 , D (3) 2 , D (4) 2 , . . . , D (3) s , D (4) s , N . (2) W rite T 1 , . . . , T u (with u = 4 s ) for the resulting sequence of disso ciated sets in (2), and set τ j := P t ∈ T j t . W e also denote X ≤ M ( T j ) := n X t ∈ S t : S ⊆ T j , | S | ≤ M o , X = M ( T j ) := n X t ∈ S t : S ⊆ T j , | S | = M o . Fix an integer K = c 2 R 1 / 2 where c 2 is a p ositive small enough constant. A prop er nonempty in terv al I ⊂ [1 , | A | ] is of T yp e II if it con tains b etw een K and | T j | − K elements of some blo c k T j . Otherwise I is of T yp e I . NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k 5 No w, set Y + j : = − X = j ( D 1 ) ∪   − δ + X = j ( D s )   and Y − j : = − X = j ( D s ) ∪   − δ + X = j ( D 1 )   for eac h 1 ⩽ j ⩽ K , and apply Prop osition 2.3. This provides orderings p of P and n of N such that the sequence p , δ, n is a sequencing and suc h that the b ounds in Prop osition 2.3 hold. W e say that an ordering t 1 , . . . , t | T 1 | of T 1 is ac c eptable if t 1 + · · · + t k / ∈ − IS( p ) ∪ ( δ + IS( n )) for all 1 ≤ k ⩽ K , and sa y that an ordering t 1 , . . . , t | T u | of T u is ac c eptable if t 1 + · · · + t k / ∈ − IS( n ) ∪ ( δ + IS( p )) for all 1 ≤ k ⩽ K . W e then state the follo wing lemma, which is an impro vemen t of Lemma 6.1 of [4]. Lemma 2.5. L et t 1 b e the or der induc e d by σ 1 on T 1 and t u b e the or der induc e d by σ s on T u . Then, we have that t 1 is ac c eptable with pr ob ability at le ast 0 . 99 and t u 0 . 99 is ac c eptable with pr ob ability at le ast 0 . 99 . Pr o of. W e prov e only the statement for t 1 since the argument for t u is identical. Let t 1 = t 1 , . . . , t | T 1 | b e our random ordering induced b y σ 1 on T 1 . By the union b ound, it suffices to sho w that P ( t 1 + · · · + t k ∈ − IS( p ) ∪ ( δ + IS( n ))) ⩽ 0 . 01 K − 1 for each 1 ⩽ k ⩽ K . Fix some 1 ⩽ k ⩽ K . Then the quantit y t 1 + · · · + t k is uniformly distributed on the set P = k ( D 1 ), which has size  | D 1 | k  . Then by Prop osition 2.3, with L := ⌊| Y + k | 1 / 2 ⌋ , we ha ve that      X = k ( D 1 ) ∩ ( − IS( p ) ∪ ( δ + IS( n )))      = O   | Y + k | 1 / 2 + X j 0 . 6 NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k A one-shot lemma. W e no w state a lemma that simultaneously con trols Type I and T ype II interv als b y sampling the splitting and the in ternal orderings in a single step. Lemma 2.7 (One-shot Type I/I I control) . Assume t ≤ exp( cK ) for a sufficiently smal l absolute c > 0 . L et D 1 , . . . , D s ⊆ Z k b e disso ciate d sets, e ach of size Θ( R ) , such that D 1 ∪ D s ∪ { δ } is disso ciate d, wher e δ := P s j =1 P d ∈ D j d . L et p and n b e se quenc es over Z k and assume that p , δ , n is a t -we ak se quencing. Fix an inte ger K = c 2 R 1 / 2 wher e c 2 is a smal l p ositive c onstant. Cho ose the sets T 1 , . . . , T u by the splitting pr o c e dur e ab ove, and then cho ose or derings t 1 , . . . , t u of the blo cks T 1 , . . . , T u as fol lows: (a) t 1 is the or der induc e d by σ 1 on T 1 and t u is the or der induc e d by σ s on T u , and we condition on the event that b oth t 1 and t u ar e acceptable with p ar ameter K (in the sense of L emma 2.5). (b) F or e ach internal adjac ent p air ( T 2 j , T 2 j +1 ) we sample uniformly at r andom the p air ( t 2 j , t 2 j +1 ) fr om the p ermissible p airs of or derings of length K (as done in [4, L emma 6.2]). L et a 1 , . . . , a | A | b e the c onc atenation a 1 , . . . , a | A | := p , t 1 , . . . , t u , n . Then: (1) (Type I I anti-concen tration) F or every T yp e II interval I ⊂ [1 , | A | ] with | I | ≤ t , P  X i ∈ I a i = 0  ≤ exp  − Θ( K log R )  . (2) (Type I anti-concen tration) F or every T yp e I interval I ⊂ [1 , | A | ] with | I | ≤ t , P  X i ∈ I a i = 0  ≤ exp  − Θ( R )  . (3) ( t -weak existence via LLL) With p ositive pr ob ability (henc e ther e exists a choic e of the r andom splittings and or derings) every interval I ⊂ [1 , | A | ] with | I | ≤ t has nonzer o sum, and ther efor e a 1 , . . . , a | A | is a t -we ak se quencing. Pr o of. The T yp e I I anti-concen tration inequality follows exactly by the same argument used in [4, Lemma 6.3]. T yp e I. Let us consider the following set of even ts: (i) F or eac h prop er nonempty interv al [ i, j ] ⊆ [ u ], | i − j | ≤ t , 0 ∈   X ≤ K ( T i − 1 ) ∪ − X ≤ K ( T i )   + τ i + · · · + τ j +   − X ≤ K ( T j ) ∪ X ≤ K ( T j +1 )   (with the con v en tion that T 0 = T u +1 = ∅ ); (ii) F or eac h 1 ≤ j ≤ min { t, u − 1 } , 0 ∈ IS t ( p ) + τ 1 + · · · + τ j +   − X ≤ K ( T j ) ∪ X ≤ K ( T j +1 )   ; and for eac h max { 2 , u − t + 1 } ≤ j ≤ u , 0 ∈ IS t ( n ) + τ u + · · · + τ j +   − X ≤ K ( T j ) ∪ X ≤ K ( T j − 1 )   . NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k 7 By the deterministic order (2), there exists ℓ suc h that each prop er nonempty in terv al J ⊆ [ u ] con tains some but not all of D (1) ℓ , . . . , D (4) ℓ ; hence P j ∈ J τ j includes a subset-sum of D ℓ of fixed size, whic h is distributed uniformly among all such subset-sums. By disso ciativity , all these sums are distinct, so eac h v alue is attained with probability at most  | D ℓ | | D ℓ | / 4  − 1 ≤ exp( − Θ( R )) thanks to Lemma 2.4. Then since | T j | = Θ( R ) and K = o ( R ) we hav e that | P ≤ K ( T j ) | = exp( o ( R )). Also note that | IS t ( p ) | , | IS n ( n ) | ≤ t = exp( o ( R )), therefore we find that conditions ( i ), ( ii ) hav e probabilit y of at most exp( − Θ( R )). This handles all Type I in terv als except the ones starting in the last K elements of t 2 j and ending in the first K elements of t 2 j +1 and the ones starting in p and ending in the first K elements of t 1 or the ones starting last K elements of t u and ending in n but these lat tw o cases are a voided since w e are conditioning on the even t that b oth t 1 and t u are acceptable and the pairs ( t 2 j , t 2 j +1 ) are p ermissible. t -we ak via LLL. Consider bad even ts E I for interv als I ⊂ [1 , | A | ], | I | ≤ t , defined by P i ∈ I a i = 0. By the previous tw o parts, P ( E I ) ≤ P with P ≤ exp( − Θ( K log R )). W e no w b ound the dep endency degree in the Lo v´ asz Lo cal Lemma. F or any interv al I ⊂ [1 , | A | ] and blo ck T i ⊆ D ℓ , the bad ev en t E I dep ends on • the random ordering of T i ; • the random splitting variables used to generate the four pieces D (1) ℓ , D (2) ℓ , D (3) ℓ , D (4) ℓ ; • the random ordering of T i ′ ⊆ D ℓ ′ where ( T i , T i ′ ) are adjacent and conditioned by the p er- missibilit y condition; • the random splitting variables used to generate the four pieces D (1) ℓ ′ , D (2) ℓ ′ , D (3) ℓ ′ , D (4) ℓ ′ . Fix E I . Since | I | ≤ t , the interv al I intersects at most t blo c ks. F or each disso ciated set D ℓ touc hed b y I , there are at most 8 asso ciated pieces, each of size Θ( R ). The num ber of interv als of length at most t that intersect a fixed blo c k of size Θ( R ) is O ( tR + t 2 ). Therefore the total num b er of bad ev en ts that can fail to b e m utually indep endent from E I is at most D = O  t ( tR + t 2 )  = O ( t 2 R + t 3 ) . Th us, b y the symmetric Lo v´ asz Lo cal Lemma, it suffices to v erify eP D ≤ 1 . F or t ≤ exp( cK ) and c > 0 sufficiently small, this holds since P ≤ exp( − Θ( K log R )) and D = exp( O (log t + log R )). Hence Pr  ∩ I E I  > 0. □ 2.4. Pro of of Theorem 1.4. W e no w complete the pro of of Theorem 1.4. Let A ⊆ Z k \ { 0 } . Apply Theorem 2.2 to obtain a dilation λ and a decomp osition λA = P ∪ N ∪ ( ∪ s j =1 D j ) with the stated prop erties. By [4, Prop osition 4.1] (see Lemma 2.3 of this pap er), we can choose orderings p of P and n of N suc h that p , δ, n is a sequencing. No w set K := c 2 R 1 / 2 = Θ  (log p ) 1 / 4  , with c 2 > 0 sufficien tly small, and apply Lemma 2.7. Part (3) of Lemma 2.7 giv es existence of a t -w eak sequencing provided t ≤ exp( cK ) for a sufficiently small absolute constan t c > 0, which yields t ≤ exp( c (log p ) 1 / 4 ) after renaming constants. Therefore, if p is large enough (i.e. p ≥ ¯ p ), with p ositive probability , the sampled ordering con tains no nontrivial zero-sum in terv al, i.e. it is a t -weak sequencing. Moreo v er we can chose c small enough so that exp  c (log p ) 1 / 4  < 2 for any prime p < ¯ p . This prov es the theorem. □ 3. Impro ved classical sequenceability (Proof of Theorem 1.3) W e explain ho w the one-shot control of T yp e I and T yp e II interv als yields the impro v ed classical b ound. Here we work in the classical (non- t -w eak) setting: we must av oid al l nontrivial interv als. 8 NEW BOUNDS FOR (WEAK) SEQUENCEABILITY IN Z k Pr o of of The or em 1.3. Let A ⊆ Z k \ { 0 } , and let us denote by p the least prime divisor of k . Apply the rectification/structure step (as in [4]) with a choice of parameters that yields disso ciated blo cks of size R = R ( A, k ) := c 1 max  (log p ) 1 / 2 , log p log | A |  . W e recall that the Structure Theorem of [4] holds for this v alue of R (the previous definition of R was functional to the weak-sequenceabilit y result). Here w e run the one-shot construction of Lemma 2.7 with K := c 2 R 1 / 2 ( c 2 > 0 sufficien tly small) . In the classical setting, w e use a union b ound o v er all nontrivial interv als I ⊂ [1 , | A | ]. There are at most | A | 2 suc h in terv als. F or T yp e I I interv als, Lemma 2.7(1) gives P  X i ∈ I a i = 0  ≤ exp  − Θ( K log R )  = exp  − Θ( R 1 / 2 log R )  . Similarly for T yp e I in terv als, Lemma 2.7(2) gives P  X i ∈ I a i = 0  ≤ exp  − Θ( R )  . Hence P ( ∃ non trivial I : X i ∈ I a i = 0) ≤ | A | 2 exp  − Θ( R 1 / 2 log R )  + | A | 2 exp  − Θ( R )  . Since | A | ≤ exp( c (log p ) 1 / 3 ) for c > 0 sufficiently small then R is at least Θ  (log p ) 2 / 3  , w e obtain that the righ t-hand side is o (1). Therefore, if p is large enough (i.e. p ≥ ¯ p ), with positive probability , the sampled ordering contains no non trivial zero-sum interv al, i.e. it is a sequencing. Moreov er we can c hose c small enough so that exp  c (log p ) 1 / 3  < 2 for any prime p < ¯ p . This pro v es the theorem. □ References [1] B. Alspac h, D. L. Kreher, and A. Pastine. The F riedlander–Gordon–Miller conjecture is true, Austr alas. J. Combin. 67 (2017), 11–24. [2] B. Alspach and G. Liversidge. On strongly sequenceable ab elian groups, Art Discr ete Appl. Math. 3 (2020), 19pp. [3] D. S. Archdeacon, J. H. Dinitz, A. Mattern, and D. R. Stinson. On partial sums in cyclic groups, J. Combin. Math. Combin. Comput. 98 (2016), 327–342. [4] B. Bedert and N. Kravitz. Graham’s rearrangement conjecture b eyond the rectification barrier, arXiv:2409.07403 . [5] M. 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(Simone Costa) DICA T AM, Universit ` a degli Studi di Brescia, Via Branze 43, I 25123 Brescia, It al y Email address : simone.costa@unibs.it (Stefano Della Fiore) DI I, Universit ` a degli Studi di Brescia, Via Branze 43, 25123 Brescia, It al y Email address : stefano.dellafiore@unibs.it