Multiplicative irreducibility of shifted multiplicative subgroups

SHIFTED MUL TIPLICA TIVE SUBGR OUPS ARE NO T RA TIO SETS SEO Y OUNG KIM, CHI HOI YIP, AND SEMIN Y OO A B S T R AC T . In a recent breakthrough, Kalmynin proved a conjecture of Le v–Sonn and a conjec- ture of S ´ ark ¨ ozy on additiv e decompositions of multiplicative subgroups of a prime field. In this paper , we prove a multiplicative analogue of Kalmynin’ s result on a generalization of the Lev– Sonn conjecture, inspired by a relev ant conjecture of S ´ ark ¨ ozy . W e sho w that all nonzero shifts of proper multiplicativ e subgroups (of size at least 3 ) are not ratio sets of the form A/ A . This in particular extends a result of Shkredov , where he showed the same for small multiplicative subgroups (of size < p 6 / 7 in F p ). W e also prov e an analogous statement over complex numbers for finite subgroups of the unit circle, which may be of independent interest. 1. I N T RO D U C T I O N Throughout the paper , let p be a prime, F p the finite field with p elements, and F ∗ p = F p \ { 0 } . For two subsets A, B of F p , we define their sumset A + B = { a + b : a ∈ A, b ∈ B } and pr oduct set AB = { ab : a ∈ A, b ∈ B } . Similarly , we can define their differ ence set A − B = { a − b : a ∈ A, b ∈ B } and ratio set A/B = { a/b : a ∈ A, b ∈ B } (provided that 0 / ∈ B ). Follo wing [ 9 , 11 ], we use R p to denote the set of nonzero squares in F p . A central theme in arithmetic combinatorics and analytic number theory is the interplay be- tween addition and multiplication. In particular , there is a substantial body of literature on additi ve and multiplicati ve decompositions of sets possessing various arithmetic structures. A celebrated conjecture in this direction is the so-called in verse Goldbach pr oblem , due to Ost- mann [ 10 ], which states that the set of primes cannot be written as a nontrivial sumset (with finitely many exceptions allo wed). The in verse Goldbach problem is still open, and we refer to the best-kno wn progress by Elsholtz–Harper [ 3 ] and Shao [ 14 ]. Similarly , Erd ˝ os conjectured that any “small perturbations” of the set of perfect squares cannot be written as a nontrivial sumset; we refer to the best-known progress due to S ´ ark ¨ ozy and Szemer ´ edi [ 13 ]. The multi- plicati ve analogues of these two questions are also well-studied. Elsholtz [ 2 ] showed that the set of shifted primes cannot be written as a nontri vial product set (with finitely many e xceptions allo wed). More recently , Hajdu–S ´ ark ¨ ozy [ 5 ] and the second author [ 24 ] studied multiplicati ve decompositions of “small perturbations” of the set of shifted k -th po wers. In this paper , we study finite field analogues of these questions. In particular , we study certain multiplicati ve decompositions of shifted multiplicative subgroups of finite fields. The following well-kno wn conjecture is due to S ´ ark ¨ ozy [ 11 ]. Conjecture 1.1 (S ´ ark ¨ ozy [ 11 ]) . Let p be a lar ge enough prime. Then R p admits no nontrivial additive decomposition, that is, ther e ar e no two subsets A, B of F p with | A | , | B | ≥ 2 , such that A + B = R p . 2020 Mathematics Subject Classification. 11P70, 11B30, 11T06. K e y words and phr ases. multiplicativ e subgroup, multiplicativ e decomposition. 1 It is natural to consider the analogue of this conjecture over multiplicativ e subgroups of finite fields. In particular , the following generalization of S ´ ark ¨ ozy’ s conjecture is well-belie ved: Conjecture 1.2. Let d ≥ 2 be fixed. Then for all suf ficiently lar ge prime power s q ≡ 1 (mo d d ) , the multiplicative subgr oup G of F q of inde x d admits no nontrivial additive decom- position. These two conjectures hav e been studied extensi vely; see for example [ 18 , 15 , 16 , 9 , 17 , 6 , 21 , 23 , 7 ]. In a breakthrough work of 2021, Hanson and Petridis [ 6 ] proved the follo wing theorem using Stepanov’ s method [ 20 ]. Theorem 1.3 (Hanson and Petridis [ 6 ]) . Let p be a prime and G be a pr oper multiplicative subgr oup of F p . If A, B are subsets of F p such that A + B ⊆ G ∪ { 0 } , then | A || B | ≤ | G | + | ( − A ) ∩ B | . T ogether with Ford’ s theorem [ 4 ] on the distribution of di visors of shifted primes, Hanson and Petridis showed that Theorem 1.3 implies an asymptotic version of Conjecture 1.1 (see [ 6 , Corollary 1.4]). In a very recent breakthrough, using a sophisticated application of Stepanov’ s method, together with v arious tools from number theory and combinatorics, Kalmynin [ 7 ] re- solved Conjecture 1.1 and made substantial progress to wards Conjecture 1.2 ov er prime fields. There are also other variants of Conjecture 1.1 and Conjecture 1.2 . For e xample, the restricted sumset variant of Conjecture 1.1 was confirmed by Shkredov [ 15 ] and the second author [ 23 ] for all finite fields F q with q odd and q > 13 . In this paper , we study a multiplicativ e analogue of the dif ference set variant of Conjecture 1.1 proposed by Le v and Sonn [ 9 ]. W e begin by recalling the Le v–Sonn conjecture. Conjecture 1.4 (Le v and Sonn [ 9 ]) . Let p be a lar ge enough prime. Then ther e is no subset A of F p such that A − A = R p ∪ { 0 } . In [ 7 ], Kalmynin used an ingenious idea to confirm the generalization of Conjecture 1.4 for multiplicati ve subgroups of prime fields. Theorem 1.5 (Kalmynin [ 7 ]) . Let p be a prime and let G be a pr oper multiplicative subgr oup of F p such that | G | / ∈ { 2 , 6 } . Then for eac h subset A of F p , we have A − A  = G ∪ { 0 } . In fact, it is implicit in his proof [ 7 , Section 3] that a slightly stronger statement holds: under the same assumption of Theorem 1.5 , if A − A ⊆ G ∪ { 0 } , then | A | 2 − | A | ≤ | G | − 1 . (1.1) Note that in the same setting, Theorem 1.3 implies the slightly weaker bound that | A | 2 − | A | ≤ | G | . Howe ver , impro ving the upper bound from | G | to | G | − 1 requires highly nontrivial efforts. As another consequence, when p ≥ 17 and p ≡ 1 (mo d 4) , inequality ( 1.1 ) slightly improves the well-known Hanson–Petridis bound √ 2 p − 1+1 2 on the clique number of the Pale y graph o ver F p [ 6 ] (that is, the largest possible size of A ⊆ F p such that A − A ⊆ R p ∪ { 0 } ) to √ 2 p − 5+1 2 . W e refer to [ 22 ] for more discussions on recent progress tow ards estimating the clique number of Pale y graphs and their generalizations. 2 Next, we turn to multiplicativ e analogues of the conjectures and results discussed above, before discussing our main result. These analogues were also initiated by S ´ ark ¨ ozy [ 12 ]. Conjecture 1.6 (S ´ ark ¨ ozy [ 12 ]) . Let p be a lar ge enough prime. Then for each λ ∈ F ∗ p , the set ( R p − λ ) \ { 0 } has no nontrivial multiplicative decomposition, that is, ther e ar e no two subsets A, B of F p with | A | , | B | ≥ 2 , such that AB = ( R p − λ ) \ { 0 } . W e note that the remov al of 0 is necessary , since if 0 ∈ R p − λ then trivially we hav e the decomposition ( R p − λ ) = { 0 , 1 } · ( R p − λ ) . In [ 8 ], inspired by Conjecture 1.2 , we formulated a generalization of this multiplicati ve S ´ ark ¨ ozy conjecture for proper multiplicati ve subgroups. Conjecture 1.7 ([ 8 , Conjecture 1.9]) . Let d ≥ 2 be fixed. Let q ≡ 1 (mo d d ) be a sufficiently lar ge prime power . Let G be the multiplicative subgr oup of F q of index d . Then for eac h λ ∈ F ∗ q , the set ( G − λ ) \ { 0 } has no nontrivial multiplicative decomposition. Using Stepanov’ s method, we made some partial progress on Conjecture 1.7 in our previous paper [ 8 , Theorem 1.11]. The following lemma is a simplified version of [ 8 , Theorem 1.1], which is the ke y ingredient in the proof of [ 8 , Theorem 1.11] in the same paper . Lemma 1.8 ([ 8 , Theorem 1.1]) . Let p be a prime and let G be a pr oper multiplicative subgr oup of F p . Let A, B ⊆ F ∗ p and λ ∈ F ∗ p . If AB + λ ⊆ G ∪ { 0 } , then | A || B | ≤ | G | + | B ∩ ( − λA − 1 ) | + | A | − 1 . Mor eover , when λ ∈ G , we have a str onger upper bound: | A || B | ≤ | G | + | B ∩ ( − λA − 1 ) | − 1 . In particular, for λ = 1 , we confirmed Conjecture 1.6 for almost all primes p [ 8 , Theorem 1.11]. W e remark that the multiplicativ e S ´ ark ¨ ozy conjecture is potentially more dif ficult than the additi ve S ´ ark ¨ ozy conjecture 1 . Indeed, in Shkredov’ s paper [ 17 ], he made important progress on Conjecture 1.2 by showing that all small multiplicativ e subgroups of prime fields hav e no nontri vial additiv e decomposition. By contrast, he obtained only a much weaker statement to ward multiplicativ e decompositions of shifted multiplicativ e subgroups: if ϵ > 0 is fixed, then for all small multiplicati ve subgroups G of F p satisfying 1 ≪ ϵ | G | ≪ ϵ p 6 / 7 − ϵ , A/ A  = ( ξ G + 1) \ { 0 } holds for all ξ ∈ F ∗ p and subsets A of F ∗ p , where ξ G = { ξ g : g ∈ G } . Similarly , when λ / ∈ G , the conclusion of Lemma 1.8 is weaker and not strong enough for our intended application to Conjecture 1.7 . Note that Shkredov’ s result confirms a multiplicati ve analogue of the generalized Lev–Sonn conjecture for small multiplicativ e subgroups. Our main result, stated below , establishes a multiplicati ve analogue of Kalmynin’ s theorem (Theorem 1.5 ), and in particular extends the 1 Priv ate communication with Ilya Shkredov . 3 abov e result of Shkredov [ 17 ] to all proper multiplicativ e subgroups of order at least 3 . Our proof is based on a combination of Stepanov’ s method and tools from symmetric polynomials. It is inspired by Kalmynin’ s paper [ 7 ], although the exact proof techniques and the choice of auxiliary polynomials are dif ferent. Theorem 1.9. Let p be an odd prime. Let G be a pr oper multiplicative subgr oup of F p with | G | ≥ 3 , and let ξ , µ ∈ F ∗ p . Then we have the following. (1) The set of nonzer o elements of ξ G + µ cannot be written as a ratio set of the form A/ A : for any A ⊆ F ∗ p , we have A/ A  = ( ξ G + µ ) \ { 0 } . (1.2) (2) The set of nonzer o elements of ( ξ G ∪ { 0 } ) + µ cannot be written as a ratio set of the form A/ A : for any A ⊆ F ∗ p , we have A/ A  =  ( ξ G ∪ { 0 } ) + µ  \ { 0 } . (1.3) W e believ e that both parts of the theorem are natural multiplicativ e analogues of Kalmynin’ s theorem (Theorem 1.5 ). Also, we remark that the assumptions that G is proper and | G | ≥ 3 are necessary (in both parts of the abov e theorem) by considering the follo wing counterexamples: • | G | = 1 : we hav e G = { 1 } and we can take A = { 1 } so that A/ A = (2 G − 1) \ { 0 } and A/ A = (( − G ∪ { 0 } ) + 1) \ { 0 } . • | G | = 2 : we hav e G = { 1 , − 1 } and we can take A = { 1 } so that A/ A = ( 1 2 G + 1 2 ) \ { 0 } . By taking A = { 1 , − 2 } , we ha ve A/ A = { 1 , − 1 / 2 , − 2 } = ( − 3 2 G ∪ { 0 } − 1 2 ) \ { 0 } . • G = F ∗ p : let g be a primitiv e root in F p and set A = { g j : 0 ≤ j ≤ p − 3 2 } , then A/ A =  g j : − p − 3 2 ≤ j ≤ p − 3 2  = F ∗ p \ {− 1 } = ( F ∗ p − 1) \ { 0 } . By taking A = F ∗ p , we simply hav e A/ A = ( F p + 1) \ { 0 } . Finally , let us consider the setting of roots of unity in the field of complex numbers. Let C be the field of complex numbers and C ∗ = C \ { 0 } . Let S 1 denote the unit circle, that is, S 1 = { z ∈ C : | z | = 1 } . In Kalmynin’ s paper [ 7 ], as well as the paper of Hanson and Petridis [ 6 ], the y also considered analogues of Theorem 1.3 and Theorem 1.5 for finite subgroups of S 1 . In a similar spirit, in the following theorem, we establish an analogue of Theorem 1.9 for finite subgroups of S 1 . Theorem 1.10. Let G be a finite subgr oup of S 1 with | G | ≥ 3 . (1) F or e very ξ , µ ∈ C ∗ and every A ⊆ C ∗ , A/ A  = ( ξ G + µ ) \ { 0 } . (2) F or e very ξ , µ ∈ C ∗ and every A ⊆ C ∗ , A/ A  =  ( ξ G ∪ { 0 } ) + µ  \ { 0 } . Similar to the remark follo wing Theorem 1.9 , the assumption | G | ≥ 3 is also necessary . 4 Organization of the paper . In Subsection 2.1 , we recall a standard result on symmetric poly- nomials. In Subsection 2.2 , we re visit a multiplicati ve analogue of the Hanson-Petridis polyno- mial introduced in our pre vious paper [ 8 ], which plays a crucial role in the proof of Lemma 1.8 . In Section 3 , we prov e Theorem 1.9 , establishing multiplicati ve analogues of the generalized Le v-Sonn conjecture. Finally , in Section 4 , we present a short proof of Theorem 1.10 . 2. P R E L I M I N A R I E S 2.1. Symmetric polynomials. In this subsection, we recall some standard families of sym- metric polynomials and fix our notation. Throughout, n ≥ 1 is an integer and x 1 , . . . , x n are indeterminates. • The power sum symmetric polynomials are defined by p k ( x 1 , x 2 , . . . , x n ) = n X i =1 x k i for k ≥ 0 . • The elementary symmetric polynomials are defined by e k ( x 1 , x 2 , . . . , x n ) = X 1 ≤ i 1 n . The following classical identity relates these two families of symmetric polynomials, which will be used in Subsection 3.2 . Lemma 2.1 (Ne wton’ s identities [ 19 ]) . F or each inte ger k ≥ 1 , we have k e k ( x 1 , x 2 , . . . , x n ) = k X i =1 ( − 1) i − 1 e k − i ( x 1 , x 2 , . . . , x n ) p i ( x 1 , x 2 , . . . , x n ) , (2.1) wher e the identity holds whenever n ≥ k ≥ 1 . 2.2. A multiplicative analogue of Hanson–Petridis polynomials. In this subsection, we re- visit a multiplicati ve analogue of the Hanson–Petridis polynomial dev eloped in our previous paper [ 8 ], and deri ve useful properties from it. Let G be a proper multiplicativ e subgroup of F p . Let A, B ⊆ F ∗ p and λ ∈ F ∗ p with | A | , | B | ≥ 2 , such that AB + λ ⊆ G ∪ { 0 } . Denote A = { a 1 , a 2 , . . . , a n } with | A | = n and B = { b 1 , b 2 , . . . , b m } with | B | = m . Let r := | B ∩ ( − λA − 1 ) | . Relabel the elements in B so that b 1 , . . . , b r ∈ B ∩ ( − λA − 1 ) and b r +1 , . . . , b m ∈ B \ ( − λA − 1 ) . Follo wing [ 8 , Section 4.1], there exists a unique solution c 1 , c 2 , . . . , c n ∈ F p of the follo wing system of equations:    P n i =1 c i a j i = 0 , 1 ≤ j ≤ n − 1 . P n i =1 c i = 1 (2.2) W e considered the following auxiliary polynomial defined as: f ( x ) = − λ n − 1 + n X i =1 c i ( a i x + λ ) n − 1+ | G | ∈ F p [ x ] . (2.3) 5 W e sho wed that f is a non-zero polynomial, each of b 1 , . . . , b r is a root of f with multiplicity at least n − 1 , and each of b r +1 , . . . , b m is a root of f with multiplicity at least n . It follo ws that r ( n − 1) + ( m − r ) n = mn − r ≤ deg f ≤ n − 1 + | G | . (2.4) Thus, when equality holds in ( 2.4 ), f must hav e degree n − 1 + | G | = mn − r and f ( x ) = C · r Y j =1 ( x − b j ) n − 1 · m Y j = r +1 ( x − b j ) n , (2.5) where C = n X i =1 c i a n − 1+ | G | i  = 0 . 3. P R O O F O F T H E O R E M 1 . 9 In this section, we prov e Theorem 1.9 . 3.1. Proof of Theorem 1.9 (1). Suppose for contradiction that there exist ξ , µ ∈ F ∗ p , a subset A ⊆ F ∗ p , and a proper multiplicati ve subgroup G of F p with order at least 3 , such that A/ A = ( ξ G + µ ) \ { 0 } . Let | A | = n . Since | G | ≥ 3 , we hav e n ≥ 2 . Since A/ A = ( ξ G + µ ) \ { 0 } , we hav e A/ ( ξ A ) = ( G + µ/ξ ) \ { 0 } . Set λ = − µ/ξ so that A/ ( ξ A ) + λ = G \ { λ } , and we follo w the same notations as in Subsection 2.2 by viewing B = 1 / ( ξ A ) . Observe first that r = | ( ξ A ) − 1 ∩ ( − λA − 1 ) | = 0 . Indeed, if we ha ve u ∈ ( ξ A ) − 1 ∩ ( − λA − 1 ) , then there exist a, a ′ ∈ A such that u = ξ − 1 a − 1 = − λ ( a ′ ) − 1 . It then follows that a ′ / ( ξ a ) = − λ ∈ A/ ( ξ A ) , and thus 0 = − λ + λ ∈ A/ ( ξ A ) + λ ⊆ G , a contradiction. Since 1 /ξ = a/ ( ξ a ) for all a ∈ A , as a trivial upper bound on A/ ( ξ A ) , we ha ve | G \ { λ }| = | A/ ( ξ A ) | ≤ n 2 − n + 1 . (3.1) Next, we deri ve an upper bound on n 2 . If λ ∈ G , Lemma 1.8 implies that n 2 = | A | 2 ≤ | G | + r − 1 = | G | − 1 , which violates inequality ( 3.1 ). Thus, we must ha ve λ / ∈ G . In this case, inequality ( 3.1 ) implies that | G | = | G \ { λ }| ≤ n 2 − n + 1 . On the other hand, by Lemma 1.8 , we get n 2 = | A | 2 ≤ | G | + r + | A | − 1 = | G | + n − 1 . Thus, we hav e n 2 − n + 1 = | G | , and equality holds in ( 2.4 ). Thus, the auxiliary polynomial f defined in equation ( 2.3 ) also satisfies equation ( 2.5 ). It follo ws that we ha ve the following equality of polynomials: f ( x ) = − λ n − 1 + n X i =1 c i ( a i x + λ ) n 2 = C · n Y j =1  x − ξ − 1 a − 1 j  n , (3.2) where C = P n i =1 c i a n − 1+ | G | i  = 0 . 6 On the middle of equation ( 3.2 ), for each 1 ≤ k ≤ n 2 , the coef ficient of x k is  n 2 k  λ n 2 − k n X i =1 c i a k i . By system ( 2.2 ), P n i =1 c i a k i = 0 for 1 ≤ k ≤ n − 1 . Thus, the coefficient of x k in f ( x ) is 0 for 1 ≤ k ≤ n − 1 . As for the right-hand side of equation ( 3.2 ), we set P ( x ) := n Y j =1 (1 − ξ a j x ) = n X t =0 ( − ξ ) t e t ( a 1 , . . . , a n ) x t . Using x − ξ − 1 a − 1 j = − ξ − 1 a − 1 j (1 − ξ a j x ) , we hav e f ( x ) = C n Y j =1 ( − 1) n ( ξ − n a − n j ) P ( x ) n , and thus the coef ficient of x k in f ( x ) is C ( − ξ ) − n 2  n Y j =1 a j  − n · [ x k ] P ( x ) n , where [ x k ] P ( x ) n denotes the coef ficient of x k in P ( x ) n . Therefore, the coef ficient of x k in P ( x ) n is 0 for 1 ≤ k ≤ n − 1 . W e now prove by induction that e k := e k ( a 1 , . . . , a n ) = 0 holds for all 1 ≤ k ≤ n − 1 . For k = 1 , the coef ficient of x in P ( x ) n is n ( − ξ ) e 1 , so we hav e n ( − ξ ) e 1 = 0 . Thus e 1 = 0 . Assume e 1 = · · · = e k − 1 = 0 for some 2 ≤ k ≤ n − 1 . Then, we have P ( x ) = 1 + ( − ξ ) k e k x k + x k +1 Q k ( x ) for some polynomial Q k ( x ) ∈ F p [ x ] . It follows that P ( x ) n = 1 + n ( − ξ ) k e k x k + x k +1 R k ( x ) . for some polynomial R k ( x ) ∈ F p [ x ] . Since the coef ficient of x k in P ( x ) n is 0 , we have n ( − ξ ) k e k = 0 , that is, e k = 0 . W e hav e thus proved that e 1 = · · · = e n − 1 = 0 . This implies that n Y i =1 ( x − a i ) = x n − e 1 x n − 1 + e 2 x n − 2 − · · · + ( − 1) n e n = x n + ( − 1) n e n . In particular , a 1 , a 2 , . . . , a n are precisely the n distinct roots of the polynomial x n − α for α := ( − 1) n − 1 e n  = 0 . It follows that A is a coset of the subgroup of n -th roots of unity in F ∗ p . Thus, we ha ve | A/ ( ξ A ) | = n . But we also have | A/ ( ξ A ) | = | G | = n 2 − n + 1 . Since n ≥ 2 , we hav e n 2 − n + 1 > n , a contradiction. This completes the proof. □ 7 3.2. Proof of Theor em 1.9 (2). Suppose for contradiction that there exist ξ , µ ∈ F ∗ p and a set A ⊆ F ∗ p such that A/ A =  ( ξ G ∪ { 0 } ) + µ  \ { 0 } . (3.3) It follo ws that | A/ A | ≥ | G | ≥ 3 , and thus we hav e n := | A | ≥ 2 . Claim 3.1. W e have n ≥ 3 . Pr oof of claim. Suppose otherwise that n = 2 . In this case, we must have | A/ A | = | G | = 3 and 0 ∈ ξ G + µ . W ithout loss of generality , write A = { 1 , c } with c ∈ F ∗ p . Then A/ A = { 1 , c, c − 1 } , and we must hav e c  = ± 1 so that c  = c − 1 . Since − µ ∈ ξ G , we have ξ G/ ( − µ ) = G . Thus, by dividing − µ on both sides of equa- tion ( 3.3 ), we obtain ( A/ A ) / ( − µ ) =  ( G ∪ { 0 } ) − 1  \ { 0 } . Let G = { 1 , ω , ω 2 } with 1 + ω + ω 2 = 0 and ω  = 1 . Then we ha ve { 1 , c, c − 1 } = { µ, µ (1 − ω ) , µ (1 − ω 2 ) } . Next, we deri ve a contradiction in each of the follo wing three cases: (1) µ = 1 . In this case we ha ve { 1 , c, c − 1 } = { 1 , 1 − ω , 1 − ω 2 } . Howe ver , (1 − ω )(1 − ω 2 ) = 1 − ω − ω 2 + 1 = 2 − ω − ω 2 = 3  = 1 . (2) 1 = µ (1 − ω ) . In this case we have { 1 , c, c − 1 } = { µ, 1 , µ (1 − ω 2 ) } . Howe ver , µ 2 (1 − ω 2 ) = µ 2 (1 − ω )(1 + ω ) = 1 + ω 1 − ω  = 1 . (3) 1 = µ (1 − ω 2 ) . In this case we have { 1 , c, c − 1 } = { µ, µ (1 − ω ) , 1 } . Ho wev er , (1 − ω )(1 + ω ) 2 = ω − ω 2  = 1 , and thus µ 2 (1 − ω ) = µ 2 (1 − ω ) µ 2 (1 − ω 2 ) 2 = 1 (1 − ω )(1 + ω ) 2  = 1 . This sho ws that n = 2 is impossible, as required. ■ Claim 3.2. W e have µ / ∈ { 1 , − 1 } . Pr oof of claim. (1) Suppose otherwise that µ = 1 . Then we have A/ A − 1 ⊆ ξ G ∪ { 0 } . W ithout loss of generality , we may assume that 1 ∈ A . Then for each a ∈ A \ { 1 } , we hav e a − 1 ∈ ξ G, 1 a − 1 = 1 − a a ∈ ξ G, thus − 1 /a ∈ G , that is, − a ∈ G . By Claim 3.1 , we hav e n ≥ 3 . Let a, b ∈ A \ { 1 } with a  = b . Then we ha ve a b − 1 = a − b b ∈ ξ G, b a − 1 = b − a a ∈ ξ G, and their ratio lies in G as follows ( a/b ) − 1 ( b/a ) − 1 = ( a − b ) /b ( b − a ) /a = − a b ∈ G. 8 On the other hand, since − a, − b ∈ G , we have a b = − a − b ∈ G . This, together with − a/b ∈ G , implies − 1 ∈ G . The above discussion shows that − A ⊆ G and thus A/ A ⊆ G . On the other hand, by equation ( 3.3 ), we hav e | A/ A | ≥ | G | . It follo ws that G =  ( ξ G ∪ { 0 } ) + 1  \ { 0 } and so − 1 ∈ ξ G . Since − 1 ∈ G , we have ξ G = G and thus G =  ( G ∪ { 0 } ) + 1  \ { 0 } . This implies that for any g ∈ G with g  = 1 , we also ha ve g − 1 ∈ G . Since − 1 ∈ G , inducti vely we hav e − 2 , − 3 , . . . , − ( p − 1) ∈ G . Thus, G = F ∗ p , violating the assumption that G is proper . (2) Suppose otherwise that µ = − 1 . W e hav e A/ A + 1 ⊆ ξ G ∪ { 0 } . By Claim 3.1 , we hav e n ≥ 3 . W ithout loss of generality , we may assume that 1 ∈ A . Then for each a ∈ A \ {− 1 } , we hav e a + 1 ∈ ξ G, 1 a + 1 = a + 1 a ∈ ξ G, and taking their ratio of these two elements in ξ G , a lies in G . Thus, A \ {− 1 } ⊆ G . No w , we want to sho w A ⊆ G (and this leads to a contradiction). Suppose otherwise that − 1 ∈ A and − 1 / ∈ G . Since n ≥ 3 , we can pick a ∈ A \ { 1 , − 1 } . Then we have a − 1 + 1 = 1 − a ∈ ξ G, − 1 a + 1 = a − 1 a ∈ ξ G. Since a ∈ G , it follo ws that 1 − a, a − 1 ∈ ξ G and thus − 1 ∈ G , a contradiction. Thus, we ha ve shown that A ⊆ G and thus A/ A ⊆ G . No w , a similar argument as in the proof of (1) sho ws that G = F ∗ p , again violating the assumption. ■ Di vide both sides of equation ( 3.3 ) by ξ and set λ := − µ ξ . Then, we hav e A/ ( ξ A ) =  ( G ∪ { 0 } ) − λ  \ { 0 } . (3.4) Let r = | ( ξ A ) − 1 ∩ ( − λA − 1 ) | . Then, r = |{ ( a, b ) ∈ A × A : a/ ( bξ ) = − λ }| , so − λ has exactly r representations in A/ ( ξ A ) . Note that if a, b ∈ A such that a/ ( ξ b ) = − λ , then b/ ( ξ a ) = − 1 / ( λξ 2 ) . Thus, − 1 / ( λξ 2 ) has e xactly r representations in A/ ( ξ A ) as well. Also, 1 /ξ has e xactly n representations in A/ ( ξ A ) . Note that − λ = µ/ξ and − 1 / ( λξ 2 ) = 1 / ( µξ ) . W e hav e µ  = ± 1 by Claim 3.2 , thus − λ, − 1 / ( λξ 2 ) , 1 /ξ are pairwise distinct. Suppose first that λ ∈ G . It then follows from equation ( 3.4 ) that | G | = | ( G ∪ { 0 } ) \ { λ }| = | A/ ( ξ A ) + λ | ≤ n 2 − ( n − 1) − 2( r − 1) = n 2 − n − 2 r + 3 . (3.5) No w , by Lemma 1.8 and inequality ( 3.5 ), we get n 2 = | A | 2 ≤ | G | + | ( ξ A ) − 1 ∩ ( − λA − 1 ) | − 1 = | G | + r − 1 ≤ n 2 − n − r + 2 , a contradiction since n ≥ 3 . Therefore, λ / ∈ G . Then equation ( 3.4 ) becomes A/ ( ξ A ) = ( G ∪ { 0 } ) − λ. (3.6) W e follow the same notations as in Subsection 2.2 by vie wing B = 1 / ( ξ A ) . 9 Note that r ≥ 1 since − λ ∈ A/ ( ξ A ) . By equation ( 3.6 ), we hav e | G | + 1 = | G ∪ { 0 }| = | A/ ( ξ A ) + λ | ≤ n 2 − n + 1 − 2( r − 1) = n 2 − n − 2 r + 3 . (3.7) Moreov er , the equality in ( 3.7 ) holds precisely when each ratio in A/ ( ξ A ) other than 1 /ξ , − λ, and − ( λξ 2 ) − 1 has exactly one representation. On the other hand, by Lemma 1.8 and inequal- ity ( 3.7 ), we get n 2 = | A | 2 ≤ | G | + | ( ξ A ) − 1 ∩ ( − λA − 1 ) | + | A | − 1 = | G | + r + n − 1 ≤ n 2 − r + 1 . Thus, we must ha ve r = 1 and n 2 − n = | G | . Moreover , we deduce that each ratio in A/ ( ξ A ) other than 1 /ξ has exactly one representation. Since n 2 − n = | G | , the equality holds in ( 2.4 ). Thus, the auxiliary polynomial f defined in equation ( 2.3 ) also satisfies equation ( 2.5 ). It follo ws that we ha ve the follo wing equality of polynomials: f ( x ) = − λ n − 1 + n X i =1 c i ( a i x + λ ) n 2 − 1 = C  x − ( ξ a 1 ) − 1  n − 1 n Y j =2  x − ( ξ a j ) − 1  n , (3.8) where C  = 0 , and by definition, a 1 is the unique element in A such that ( ξ a 1 ) − 1 ∈ ( ξ A ) − 1 ∩ ( − λA − 1 ) , that is, µa 1 ∈ A . By replacing the set A with A/a 1 and repeating the abov e argument, we may , without loss of generality , assume that a 1 = 1 . Then we must hav e µ ∈ A . Recall that system ( 2.2 ) states that P n i =1 c i = 1 and P n i =1 c i a k i = 0 for 1 ≤ k ≤ n − 1 . Since | G | = n 2 − n and λ / ∈ G , it follows that f (0) = − λ n − 1 + λ n 2 − 1 n X i =1 c i = − λ n − 1 + λ n 2 − 1 = λ n − 1 ( λ n 2 − n − 1)  = 0 . On the other hand, a similar computation shows that the coef ficient of x k in f ( x ) is 0 for 1 ≤ k ≤ n − 1 . Thus, there exists R ( x ) ∈ F p [ x ] such that f ( x ) = f (0) + x n R ( x ) . It follo ws that the coef ficient of x k in f ′ ( x ) is 0 for 0 ≤ k ≤ n − 2 . On the other hand, differentiating ( 3.8 ) and di viding it by f , we obtain f ′ ( x ) f ( x ) = n − 1 x − ξ − 1 + n X j =2 n x − ( ξ a j ) − 1 . Using the identity 1 1 − z = 1 + z + z 2 + · · · + z n − 2 + z n − 1 1 − z , we hav e f ′ ( x ) f ( x ) = − n − 2 X k =0  ( n − 1) ξ k +1 + n n X j =2 ( ξ a j ) k +1  x k + x n − 1 · U ( x ) , (3.9) for some rational function U ( x ) such that U ( x ) f ( x ) is a polynomial. 10 Multiplying both hand-sides of equation ( 3.9 ) by f ( x ) = f (0) + x n R ( x ) and using the facts that f ′ ( x ) is divisible by x n − 1 and f (0)  = 0 , we obtain that for each 0 ≤ k ≤ n − 2 , ( n − 1) ξ k +1 + n n X j =2 ( ξ a j ) k +1 = 0 . Equi valently , for 1 ≤ k ≤ n − 1 , n X j =2 a k j = − n − 1 n . It follows that p k ( a 2 , a 3 , . . . , a n ) is uniquely determined for 1 ≤ k ≤ n − 1 . Newton’ s identities (Lemma 2.1 ) then allo w us to compute e k ( a 2 , a 3 , . . . , a n ) for 1 ≤ k ≤ n − 1 , and thus the set { a 2 , a 3 , . . . , a n } is uniquely determined, say it is giv en by A ∗ . As a summary , we ha ve sho wn the following claim. Claim 3.3. If { 1 , µ } ⊆ A and A satisfies equation ( 3.6 ) , then A = { 1 } ∪ A ∗ , wher e A ∗ is the set defined above, and µ ∈ A ∗ . Moreo ver , each ratio in A/ ( ξ A ) other than 1 /ξ has exactly one r epresentation. No w , set e A = µ/ A . Note that we still have e A ξ e A = A ξ A = ( G ∪ { 0 } ) − λ. Moreov er , since 1 , µ ∈ A , we also ha ve 1 , µ ∈ e A . Thus, we can apply the abo ve claim to e A to conclude that e A = { 1 } ∪ A ∗ = A . Since n ≥ 3 , we can take a ′ ∈ A \ { 1 , µ } . Then, we have µ/a ′ ∈ A and µ/a ′  = 1 . The two ordered pairs (1 , a ′ ) and  µ a ′ , µ  are distinct and both lie in A × A . The y giv e the same ratio in A/ ( ξ A ) as 1 ξ a ′ = µ/a ′ ξ µ . This sho ws that the ratio 1 / ( ξ a ′ ) (which is not 1 /ξ ) has at least two representations in A/ ( ξ A ) , a contradiction. □ 4. T H E C O M P L E X S E T T I N G W e first revie w some basic facts on M ¨ obius transformations. Let b C = C ∪ {∞} be the Riemann sphere. A M ¨ obius transformation is a map M : b C → b C , M ( z ) = az + b cz + d , where a, b, c, d ∈ C and ad − bc  = 0 . W e interpret the v alue at ∞ by M ( ∞ ) = ( a/c, c  = 0 , ∞ , c = 0 , and M  − d c  = ∞ ( c  = 0) . Every M ¨ obius transformation is bijecti ve on b C and its in v erse map is also a M ¨ obius transforma- tion. In particular , the in version map ι ( z ) := 1 /z 11 is a M ¨ obius transformation. W e will use the following standard facts (see, for example, [ 1 , Ch. 3]): • M ¨ obius transformations send generalized circles (circles/lines in b C ) to generalized cir- cles. • M ¨ obius transformations are uniquely determined by their v alues on three distinct points. • T wo distinct generalized circles meet in at most two points. The follo wing lemma will be useful in the proof of Theorem 1.10 . Lemma 4.1. Let m ≥ 3 and let G = { ω k : 0 ≤ k < m } ⊆ S 1 be the gr oup of m -th r oots of unity , wher e ω = e 2 π i/m . If a M ¨ obius transformation ψ satisfies ψ ( S 1 ) = S 1 and ψ ( G ) = G , then ther e exists ζ ∈ G such that ψ ( z ) = ζ z for all z ∈ b C or ψ ( z ) = ζ /z for all z ∈ b C . Pr oof. Note that ψ ( S 1 ) = S 1 and ψ is continuous on S 1 . Since ψ permutes the elements in the set G , it has to preserve or rev erse the cyclic order on the m vertices of the regular m -gon G . Thus, ψ restricted to G is either a rotation z 7→ ζ z or a reflection z 7→ ζ /z for some ζ ∈ G . Since M ¨ obius maps are determined by their values on three distinct points, ψ must agree with that rotation/reflection on all of b C . □ W e end the paper by presenting a proof of Theorem 1.10 . Pr oof of Theor em 1.10 . Write m := | G | ≥ 3 . Let ξ , µ ∈ C ∗ , and set V := µ + ξ G, C := µ + ξ S 1 , where C is the Euclidean circle of center µ and radius | ξ | . Let T ( z ) := ( z − µ ) /ξ for all z ∈ b C , so that T ( C ) = S 1 and T ( V ) = G . Define ψ := T ◦ ι ◦ T − 1 . (1) Suppose for contradiction that there exists A ⊆ C ∗ such that A/ A = X := ( µ + ξ G ) \ { 0 } . Since A/ A is closed under the in version map ι : x 7→ x − 1 , we hav e ι ( X ) = X and 1 ∈ X . Next, we consider the follo wing two cases. Case 1: 0 / ∈ V . Then X = V and thus ι ( V ) = V . Since V contains m ≥ 3 points of the circle C , the generalized circles C and ι ( C ) meet in at least three points. It follows that ι ( C ) = C since two distinct generalized circles meet in at most two points, and thus ψ ( S 1 ) = S 1 . Moreov er , for each g ∈ G , we have v := T − 1 ( g ) = µ + ξ g ∈ V , hence ι ( v ) ∈ V , so ψ ( g ) = T ( ι ( v )) ∈ T ( V ) = G. This, together with the injecti vity of ψ , shows that ψ ( G ) = G . By Lemma 4.1 , we hav e ψ ( ∞ ) ∈ {∞ , 0 } . Howe ver , a direct computation giv es ψ ( ∞ ) = T  ι ( T − 1 ( ∞ ))  = T ( ι ( ∞ )) = T (0) = − µ/ξ ∈ C ∗ , (4.1) a contradiction. Case 2: 0 ∈ V . Then X = V \ { 0 } has size m − 1 . 12 W e first claim m ≥ 4 . Assume otherwise m = 3 . Then | X | = 2 . Since ι ( X ) = X , it follo ws that X = {− 1 , 1 } . Write G = { 1 , ω , ω 2 } with ω = e 2 π i/ 3 . Then we must have {− 1 , 0 , 1 } = { µ + ξ , µ + ξ ω , µ + ξ ω 2 } . It follo ws that 0 = ( µ + ξ ) + ( µ + ξ ω ) + ( µ + ξ ω 2 ) = 3 µ , that is, µ = 0 , a contradiction. Let g 0 ∈ G be the unique element with µ + ξ g 0 = 0 . Since 0 ∈ V ⊆ C , the image ι ( C ) is a line. It follows that ψ ( S 1 ) is a line. For each g ∈ G \ { g 0 } , we hav e µ + ξ g ∈ X , so ι ( µ + ξ g ) ∈ X and ψ ( g ) = T  ι ( µ + ξ g )  ∈ T ( X ) = G \ { g 0 } ⊆ S 1 . Thus, since ψ is injective, ψ ( G \ { g 0 } ) is a set of m − 1 ≥ 3 distinct points contained in the intersection of the line ψ ( S 1 ) with S 1 , contradicting the fact that a line meets a circle in at most two points. (2) Suppose for contradiction that there exists A ⊆ C ∗ such that A/ A = X :=  ( ξ G ∪ { 0 } ) + µ  \ { 0 } . Since A/ A is closed under in version, we ha ve ι ( X ) = X and 1 ∈ X . W e first claim that m ≥ 4 . Suppose otherwise that m = 3 . W ithout loss of generality , assume that 1 ∈ A . Note that we can find an element r ∈ A \ { 1 , − 1 } for otherwise A/ A ⊆ { 1 , − 1 } . It follows that A/ A = { 1 , r, r − 1 } . W e can then follow the proof of Claim 3.1 verbatim to get a contradiction. Next, we consider the follo wing two cases. Case 1: 0 / ∈ V . Then X = V ∪ { µ } . For each v ∈ V we hav e ι ( v ) ∈ X . Note that at most one v ∈ V satisfies ι ( v ) = µ (namely v = 1 /µ ), so for at least m − 1 ≥ 3 distinct v ertices v ∈ V , we hav e ι ( v ) ∈ V ⊆ C . Hence, the generalized circles C and ι ( C ) meet in at least three points. It follo ws that ι ( C ) = C and thus ψ ( S 1 ) = S 1 . Moreov er , since 0 / ∈ V , we hav e T ( X ) = G ∪ { 0 } . For each g ∈ G , µ + ξ g ∈ X implies ψ ( g ) = T  ι ( µ + ξ g )  ∈ T ( X ) = G ∪ { 0 } , and because ψ ( g ) ∈ S 1 we get ψ ( g ) ∈ G . Thus ψ ( G ) = G . By Lemma 4.1 , ψ ( ∞ ) ∈ {∞ , 0 } . Howe ver , as in equation ( 4.1 ), we hav e ψ ( ∞ ) = − µ/ξ ∈ C ∗ , a contradiction. Case 2: 0 ∈ V . Let g 0 ∈ G be the unique element with µ + ξ g 0 = 0 . First, we claim m ≥ 5 . Assume otherwise that G = { 1 , − 1 , i, − i } . Note that for g ∈ G , we hav e µ + ξ g = µ − µg − 1 0 g = µ (1 − g − 1 0 g ) . Thus, µ + ξ G = µ (1 − G ) . It follo ws that X = µ { 1 , 2 , 1 − i, 1 + i } , ι ( X ) = µ − 1  1 , 1 2 , 1 + i 2 , 1 − i 2  = 1 2 µ 2 X . Since X = ι ( X ) , we must hav e 2 µ 2 = 1 . On the other hand, 1 ∈ X implies that µ ∈ { 1 , 1 2 , 1+ i 2 , 1 − i 2 } , a contradiction. Since 0 ∈ V , we hav e 0 ∈ C and thus ι ( C ) is a line. It follo ws that ψ ( S 1 ) is a line. One checks that X = ( V \ { 0 } ) ∪ { µ } , T ( V \ { 0 } ) = G \ { g 0 } , and T ( µ ) = 0 , thus T ( X ) =  G \ { g 0 }  ∪ { 0 } . For each g ∈ G \ { g 0 } , we hav e ψ ( g ) ∈ ψ ( S 1 ) ; on the other hand, µ + ξ g ∈ X , hence ι ( µ + ξ g ) ∈ X and ψ ( g ) = T  ι ( µ + ξ g )  ∈ T ( X ) ⊆ S 1 ∪ { 0 } . 13 Since ψ is injecti ve, it follo ws that there are m − 2 ≥ 3 distinct points contained in the intersec- tion of a line with S 1 , a contradiction. □ A C K N O W L E D G M E N T S S. Kim and C.H. Y ip thank Institute for Basic Science for hospitality on their visit, where part of this project was discussed. C.H. Y ip also thanks Swaroop Hegde, Gior gis Petridis, and Ilya Shkredov for helpful discussions. C.H. Y ip was supported in part by an NSERC fellowship. S. Y oo was supported by the Institute for Basic Science (IBS-R029-C1). R E F E R E N C E S [1] L. V . Ahlfors. Complex analysis . International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New Y ork, third edition, 1978. [2] C. Elsholtz. Multiplicativ e decomposability of shifted sets. Bull. Lond. Math. Soc. , 40(1):97–107, 2008. [3] C. Elsholtz and A. J. Harper . Additiv e decompositions of sets with restricted prime factors. T rans. Amer . Math. Soc. , 367(10):7403–7427, 2015. [4] K. Ford. The distribution of integers with a divisor in a giv en interval. Ann. of Math. (2) , 168(2):367–433, 2008. [5] L. Hajdu and A. S ´ ark ¨ ozy . On multiplicativ e decompositions of polynomial sequences, III. Acta Arith. , 193(2):193–216, 2020. [6] B. Hanson and G. Petridis. Refined estimates concerning sumsets contained in the roots of unity . Pr oc. Lond. Math. Soc. (3) , 122(3):353–358, 2021. [7] A. Kalmynin. On additiv e irreducibility of multiplicativ e subgroups, 2025. [8] S. Kim, C. H. Y ip, and S. Y oo. Multiplicativ e structure of shifted multiplicativ e subgroups and its applications to Diophantine tuples. Canad. J. Math., to appear . [9] V . F . Lev and J. Sonn. Quadratic residues and dif ference sets. Q. J. Math. , 68(1):79–95, 2017. [10] H.-H. Ostmann. Additive Zahlentheorie. Erster Teil: Allgemeine Untersuchung en. Zweiter Teil: Spezielle Zahlenmengen , volume 11 of Er gebnisse der Mathematik und ihr er Grenzgebiete , (N.F .), Hefte 7 . Springer- V erlag, Berlin-G ¨ ottingen-Heidelberg, 1956. [11] A. S ´ ark ¨ ozy . On additi ve decompositions of the set of quadratic residues modulo p . Acta Arith. , 155(1):41–51, 2012. [12] A. S ´ ark ¨ ozy . On multiplicativ e decompositions of the set of the shifted quadratic residues modulo p . In Number theory , analysis, and combinatorics , De Gruyter Proc. Math., pages 295–307. De Gruyter , Berlin, 2014. [13] A. S ´ ark ¨ ozy and E. Szemer ´ edi. On the sequence of squares. Mat. Lapok , 16:76–85, 1965. [14] X. Shao. On an in verse ternary Goldbach problem. Amer . J . Math. , 138(5):1167–1191, 2016. [15] I. D. Shkredov . Sumsets in quadratic residues. Acta Arith. , 164(3):221–243, 2014. [16] I. D. Shkredov . Dif ference sets are not multiplicativ ely closed. Discr ete Anal. , pages Paper No. 17, 21, 2016. [17] I. D. Shkredov . Any small multiplicati ve subgroup is not a sumset. F inite Fields Appl. , 63:101645, 15, 2020. [18] I. E. Shparlinski. Additi ve decompositions of subgroups of finite fields. SIAM J . Discr ete Math. , 27(4):1870– 1879, 2013. [19] R. P . Stanle y . Enumerative combinatorics. Vol. 2 , volume 62 of Cambridge Studies in Advanced Mathematics . Cambridge Univ ersity Press, Cambridge, 1999. [20] S. A. Stepanov . On the number of points of a hyperelliptic curve over a finite prime field. Izv . Akad. Nauk SSSR, Ser . Mat. , 33:1171–1181, 1969. [21] C. H. Y ip. Additive decompositions of lar ge multiplicativ e subgroups in finite fields. Acta Arith. , 213(2):97– 116, 2024. [22] C. H. Y ip. Exact v alues and improved bounds on the clique number of cyclotomic graphs. Des. Codes Cryp- togr . , 93(12):5131–5142, 2025. [23] C. H. Y ip. Restricted sumsets in multiplicative subgroups, 2025. Canad. J. Math., to appear . [24] C. H. Y ip. Multiplicativ e irreducibility of small perturbations of the set of shifted k -th powers. Combinator- ica , 46(1):Paper No. 1, 2026. 14 D E PA RT E M E N T M A T H E M A T I K U N D I N F O R M A T I K , U N I V E R S I T ¨ A T B A S E L , S P I E G E L G A S S E 1 , 4 0 5 1 B A S E L , S W I T Z E R L A N D Email addr ess : seoyoung.kim@unibas.ch S C H O O L O F M A T H E M AT I C S , G E O R G I A I N S T I T U T E O F T E C H N O L O G Y , G A 3 0 3 3 2 , U N I T E D S TA T E S Email addr ess : cyip30@gatech.edu D I S C R E T E M A T H E M A T I C S G R O U P , I N S T I T U T E F O R B A S I C S C I E N C E , 5 5 E X P O - R O Y U S E O N G - G U , D A E - J E O N 3 4 1 2 6 , S O U T H K O R E A Email addr ess : syoo19@ibs.re.kr 15