aa r X i v : . [ m a t h . M G ] D ec LARGE SIGNED SUBSET SUMS
GERGELY AMBRUS AND BERNARDO GONZ ´ALEZ MERINO
Abstract.
We study the following question: for given d > n > d and k n , what is thelargest value c ( d, n, k ) such that from any set of n unit vectors in R d , we may select k vectors withcorresponding signs ± c ( d, n, k )?The problem is dual to classical vector sum minimization and balancing questions, which havebeen studied for over a century. We give asymptotically sharp estimates for c ( d, n, k ) in the generalcase. In several special cases, we provide stronger estimates: the quantity c ( d, n, n ) corresponds tothe ℓ p -polarization problem, while determining c ( d, n,
2) is equivalent to estimating the coherenceof a vector system, which is a special case of p -frame energies. Two new proofs are presented forthe classical Welch bound when n = d + 1. For large values of n , volumetric estimates are appliedfor obtaining fine estimates on c ( d, n, c (2 , n, k ) aregiven. Finally, we determine the exact value of c ( d, d + 1 , d + 1) under some extra assumptions. History and results
The study of vector sum problems dates back more than a century: see e.g. the 1913 work ofSteinitz [30] answering a question of Riemann and L´evy. In the present article, we will consider thedual of two classical problems belonging to this family.The unit vector balancing problem asks for the following. Given unit vectors u , . . . , u n in R d ,one should find signs ε , . . . , ε n ∈ {± } so that the sum ε u + . . . + ε n u n has small norm. Equivalently, the goal is to partition the vectors into two classes so that thecorresponding partial sums are close to each other. It is natural to look for the best possiblebound: in 1963, Dvoretzky [22] asked for determining(1.1) max ( u i ) n ⊂ S d − min ε ∈{± } n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 ε i u i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for fixed d and n > d , where S d − denotes the unit sphere of R d . Here and later on, | . | stands forthe Euclidean (or ℓ ) norm. Spencer [28] gave a probabilistic proof showing that for every n > d ,the above quantity equals to √ d (note that this bound is independent of the number of vectors!).Moreover, the same bound holds for sets of vectors of norm at most
1. Sharpness is illustrated forexample by taking n = d and setting the vector set ( u i ) d to be an orthonormal base of R d . Relatedcombinatorial games were studied by Spencer [27]. Generalizing Dvoretzky’s question, B´ar´any andGrinberg [10] showed that given any set of vectors in the unit ball of a d -dimensional normed space, one may always find corresponding signs so that the signed sum has norm at most d .Switching millennia did not halt related research: Swanepoel [31] showed that given an odd number of unit vectors in a normed plane, there exists a corresponding signed sum of norm at most1. Blokhuis and Chen [13] considered yet another twist of the problem, allowing for the coefficients Mathematics Subject Classification.
Key words and phrases.
Vector sums, sign sequences, coherence bounds, polarization problems, extremal vectorsystems.Research of the first author was supported by NKFIH grants PD-125502 and KKP-133819. This research is a resultof the activity developed within the framework of the Programme in Support of Excellence Groups of the Regi´on deMurcia, Spain, by Fundaci´on S´eneca, Science and Technology Agency of the Regi´on de Murcia. The second author ispartially supported by Fundaci´on S´eneca project 19901/GERM/15 and by MICINN Project PGC2018-094215-B-I00,Spain. o be 0 as well. Finally, we list another variation of (1.1) due to Koml´os [29], which has become acentral question in geometric discrepancy theory. His conjecture states that there exists a uniformconstant c so that for an arbitrary set u , . . . , u n of (Euclidean) unit vectors in R d , one may selecta sequence of signs ε , . . . , ε n ∈ {± } so that k ε u + . . . + ε n u n k ∞ c. The currently strongest upper bound is due to Banaszczyk [7], who proved that a sum of O ( √ log d ) ℓ ∞ -norm may always be found. Further vector balancing problems are listed in the survey ofB´ar´any [9].The other question which lies at the centre of our attention is that of subset sums . Given a setof vectors ω n = { u , . . . , u n } ⊂ R d of norm at most 1 which sum to 0, and a fixed k n , the goal isto find a subset U k of ω n of cardinality k , so that the sum of the vectors in U k has small norm. Itfollows from the result of Steinitz [30] that there always exists a subset of size k whose sum has normat most d . In an article of the first named author written jointly with B´ar´any and Grinberg [2], itis proven that for general norms, the upper bound of ⌊ d/ ⌋ holds for arbitrary k n , whereas forthe Euclidean norm, the optimal upper bound is of the order of magnitude Θ( √ d ). Swanepoel [32]considered sets of unit vectors in general Banach spaces, all of whose k -element subset sums havesmall norm.In all the above questions, the primary task is to find sums with small norm . In the presentpaper, we turn our interest to the reverse direction: we set the goal to find signed sums which are large . For doing so, we will assume that all vectors of the family in question are of norm 1, since nonontrivial bound may hold for vectors taken from the unit ball. Unifying the two main questionsabove, we are going to look for large signed subset sums . When the size of the sought-after subsetequals to the number of the vectors, we reach the reverse of the original unit vector balancingproblem.We only consider the Euclidean case; analogous questions for general norms may be subject tofuture research.Accordingly, let us introduce the following notion. Definition 1.
For any d > , n > and k n , let c ( d, n, k ) be the largest value so that forevery set of unit vectors ω n = { u , . . . , u n } ∈ S d − there exist indices i < · · · < i k n andcorresponding signs ε , . . . , ε k ∈ {± } such that (1.2) (cid:12)(cid:12) ε u i + . . . + ε k u i k (cid:12)(cid:12) > c ( d, n, k ) . Equivalently, c ( d, n, k ) = min max | P kj =1 ε j u i j | , where minimum is taken over all n -elementvector sets ω n = { u , . . . , u n } ⊂ S d − , while maximum is taken over all k -element subsets of ω n and all sign sequences ε ∈ {± } n . That c ( d, n, k ) exists follows by a usual compactness argument.Our primary interest is to estimate the quantities c ( d, n, k ), and determine their exact value,whenever possible.To begin with, we note that the triangle inequality implies the trivial upper bound(1.3) c ( d, n, k ) k, which may only be sharp if k = 1 or d = 1. Another simple observation is that if 1 k < k n ,then(1.4) c ( d, n, k ) c ( d, n, k )since given a maximal k -term sum, any additional vector may be oriented in a way so that adding itdoes increase the norm of the sum. Moreover, if 2 d < d , then embedding R d into R d impliesthat c ( d , n, k ) c ( d , n, k ). Obviously, we also have c ( d, n , k ) c ( d, n , k ) for 1 n < n .To obtain the simplest lower bound, we turn to an old tool. Signed sums of maximal norm appearin the argument of Bang [8] used for the solution of Tarski’s plank problem [33]. The statementbelow, known as Bang’s Lemma, is presented in the following form in [5]. roposition 1 (Bang) . If u , . . . , u n are unit vectors in R d , and the signs ε , . . . , ε n ∈ {± } arechosen so as to maximize the norm | P n ε i u i | , then (1.5) h ε i u i , n X ε j u j i > holds for every i . Summing (1.5) over i = 1 , . . . , n , we readily obtain that(1.6) (cid:12)(cid:12)(cid:12) n X ε i u i (cid:12)(cid:12)(cid:12) > √ n. The same estimate may alternatively be shown by taking the average over all sign sequences:12 n X ε ∈{± } n (cid:12)(cid:12)(cid:12) n X ε i u i (cid:12)(cid:12)(cid:12) = n X | u i | = n, since all the mixed terms h ε i u i , ε j u j i for i = j cancel. Moreover, the argument also shows that(1.6) is sharp if and only of the vector system ω n is orthonormal.Applying (1.6) to any subset of ω n of cardinality k , we reach the first nontrivial lower bound on c ( d, n, k ): Proposition 2.
For arbitrary d, n > and k n , c ( d, n, k ) > √ k holds. This estimate issharp if and only if n d . Indeed, for the estimate being sharp, we need every k -element subset of ω n to be orthonormal,which implies that ω n is orthonormal itself.Among other results, we are going to give asymptotic estimates for c ( d, n, k ). To formulate these,we will use the standard asymptotic notations O ( . ) , o ( . ) , Ω( . ) , Θ( . ) as defined in [25].In the special case k = n , using the connection with the p = 1 case of the ℓ p -polarization problemon the sphere S d − , the sharp asymptotic bound was proved in [1]: Theorem 1 (A., Nietert [1], Theorem 4) . If d, n → ∞ along with n > d , then c ( d, n, n ) = Θ (cid:18) n √ d (cid:19) . Therefore, the order of magnitude of c ( d, n, n ) proves to be Θ( n ) as opposed to the lower boundΩ( √ n ) provided by Proposition 2.The next result determines the order of magnitude of c ( d, n, k ) in the general case. Theorem 2.
For arbitrary d > , n > d and k > , (1.7) c ( d, n, k ) > max ( k − k d +1 d − n − d − , r π · √ d · k ) and (1.8) c ( d, n, k ) k − α · d · k d +1 d − n − d − for k < · d − k − α · k d +1 d − n − d − for · d − k n if n is sufficiently large, where α , α > are absolute constants. Furthermore, for every k > √ d e − d · n , (1.9) c ( d, n, k ) ϕ √ π · √ d · k olds with ϕ = r W (cid:16) n k (cid:17) , where W is the principal branch of the Lambert W function; equiva-lently, ϕ is the positive solution of the equation (1.10) ϕ · kn = e − ϕ . Determining the extremal vector systems ω n may only be hoped for in a few special cases. Wefirst discuss the case k = 2. Since(1.11) | u + v | = 2 + 2 h u, v i for unit vectors u, v ∈ S d − , bounding c ( d, n,
2) is equivalent to estimating max i Welch bound [34] states that one can always select i = j so that(1.12) |h u i , u j i| > s n − dd ( n − d + 2 n < d + O ( d ). For the n = d + 1case, (1.11) and (1.12) imply that c ( d, d + 1 , > p /d . We provide two new proofs for thisestimate: Theorem 3. For every d > , (1.13) c ( d, d + 1 , 2) = r d . The sharp bound is attained if and only if, up to sign changes, the vector system ω d +1 forms thevertex set of a regular d -dimensional simplex inscribed in S d − . We are also interested in the other end of the spectrum, when n is much larger than d . In thiscase, the coherence bounds of Welch and Bukh-Cox are not sharp. Instead, we may turn to the spherical cap packing problem , which asks for finding the largest radius R d,n so that n sphericalcaps of radius R d,n may be packed in S d − . For a detailed survey of that question, see Section2.6 of [15]. Estimating c ( d, n, 2) is equivalent to bounding the packing density of non-overlappingpairs of antipodal spherical caps of equal size. Using volumetric estimates, we prove the followingbounds for large values of n : Theorem 4. For each sufficiently large d , there exists an N > , so that for every n > N , (1.14) 2 − . n − d − < c ( d, n, < − . n − d − . Next, we study the problem in the plane. Unlike in most of the higher dimensional cases, herewe are able to derive sharp bounds. Theorem 5. The following lower bounds hold in the plane: (1.15) c (2 , n, k ) > k cos ( k − π n for even values of k , and (1.16) c (2 , n, k ) > r k − k + 1) cos ( k − π n when k is odd. These bounds are sharp if and only if n is divisible by ( k − . In those cases, equalityis attained if and only if {± u , . . . , ± u n } forms the vertex set of a regular nk − -gon inscribed in S ,each vertex taken with multiplicity k − . e finish our discussion with two further special cases. First, when k = n = d , Proposition 2shows that extremizers are exactly the orthonormal systems. Second, for k = n = d + 1, naturalintuition and numerical experiments suggest that each extremal configuration is, up to sign changes,the union of the vertex set of an even dimensional regular simplex and an orthonormal basis of theorthogonal complement of its subspace. The following conjecture, already proven for d = 2 in [19,Thm. 1.3], is from [19] and [26]. Notice that c ( d, n, n ) was considered in [19] as an application tounderstand the behavior of the circumradius with respect to the Minkowski addition of n centrallysymmetric sets in R d . Conjecture 1. For any d > , c ( d, d + 1 , d + 1) = √ d + 2 . The sharp bound is realized if andonly if, up to sign changes, ω d +1 is the union of the vertex set of a regular simplex in a subspace H centered at the origin, and an orthonormal basis of H ⊥ , where H is an even dimensional linearsubspace of R d . We conclude the article with a proof of this bound under special assumptions: Theorem 6. Assume that d is even, and the unit vectors u , . . . , u d +1 ∈ S d − satisfy P d +1 i =1 u i = 0 .Then there exist signs ε , . . . , ε d +1 ∈ {± } so that (1.17) | ε u + . . . + ε d +1 u d +1 | > √ d + 2 . General asymptotic estimates In this section we are going to use the notions of δ -nets and δ -separated sets in S d − . Bydistance we will mean the spherical (geodesic) distance on S d − . Also, for x ∈ S d − and r ∈ [0 , π ],let C ( x, r ) denote the spherical cap of S d − with centre x and radius r . The volume of the unitball B d , denoted as κ d , is given by κ d = π d/ Γ( d + 1)(see e.g. [4]). The surface area of S d − is d κ d . Let σ stand for the normalized surface area measureon S d − (thus, σ ( S d − ) = 1).Let C r denote a spherical cap of S d − of radius r : that is, C r = C ( x, r ) for an arbitrary x ∈ S d − .By projecting a cap radially to its boundary hyperplane and to the tangent hyperplane at its center,respectively, we derive the simple estimates(2.1) κ d − d κ d sin d − r σ ( C r ) κ d − d κ d tan d − r. We note for later reference that Gautschi’s inequality implies that(2.2) 1 √ πd < κ d − d κ d = Γ( d + 1) d √ π Γ( d +12 ) < √ π √ d + 2 d for every d .As usual, a set X ⊂ S d − is called a δ -net , if for arbitrary y ∈ S d − there exists x ∈ X so thatthe spherical distance between x and y is at most δ . Equivalently, the spherical caps of radii δ centred at the points of X completely cover S d − . A set Y ⊂ S d − is δ -separated , if the distancebetween any two of its points is at least δ – equivalently, the spherical caps of radius δ/ Y are pairwise non-overlapping. Furthermore, Y is a maximal δ -separated set ,if appending any point of S d − \ Y to it results in losing δ -separatedness. It is well known thatmaximal δ -separated sets are (minimal) δ -nets. Moreover, if X ⊂ S d − is a maximal δ -separatedset, then | X | = Θ( δ − ( d − ) . This is implied by e.g. Theorem 6.3.1 of [15], which states that if X isa maximal δ -separated set with δ < π/ 2, then(2.3) √ π sin − ( d − δ < | X | < 23 ( d − / sin − ( d − δ · − ( d − / . roof of Theorem 2. First, we set off for the lower bound. We consider two cases depending onwhether k = Θ( n ) or k = o ( n ). Note that the two terms in the estimate (1.7) are equal when k = − r πd ! d − n ≈ e − q d π n ;for smaller values of k , the first term dominates, while for larger k ’s, the second term is larger.Throughout the proof, ω n = { u , . . . , u n } ⊂ S d − will be an arbitrary n -element unit vector set.We will also use the notation ± ω n = { u , − u , . . . , u n , − u n } . When k is large, we apply the method of [1] for proving Theorem 1. Take k vectors of ω n arbitrarily, say, u , . . . , u k . Then (see Proposition 3 of [1])(2.4) max ε ∈{± } k (cid:12)(cid:12)(cid:12) k X i =1 ε i u i (cid:12)(cid:12)(cid:12) = max v ∈ S d − k X i =1 |h v, u i i| . The quantity on the right hand side is easy to estimate:max v ∈ S d − k X i =1 |h v, u i i| > Z S d − k X i =1 |h v, u i i| d σ ( v )= k Z S d − |h v, u i| d σ ( v )= k dκ d Z t (1 − t ) d − ( d − κ d − d t = k κ d − dκ d > k r πd , which is the second term of the estimate in (1.7).Next, we establish the estimate for small k ’s. Let r so that σ ( C r ) = k n . By (2.1) and (2.2), 1 √ πd sin d − r < k n . Using that on [0 , π/ π x sin x , this implies that r < (cid:16) kn (cid:17) d − . Accordingly, for any x ∈ S d − and for every y ∈ C ( x, r ), we have(2.5) h x, y i > cos r > − (cid:16) kn (cid:17) d − . Let us denote by X ) the cardinality of a finite set X ⊂ R d . Since Z S d − ± ω n ) ∩ C ( x, r )) d σ ( x ) = X u ∈± ω n σ ( { x ∈ S d − : u ∈ C ( x, r ) } ) = 2 n · k n = k, there exists some x ∈ S d − for which at least k vectors of ± ω n lie in C ( x, r ). Let v , . . . , v k ∈ ± ω n be k such vectors. Then, by (2.5),(2.6) | v + . . . + v k | > k X i =1 h x, v i i > k − k d +1 d − n − d − , hich is the first term of desired lower bound.Now we turn to the upper bounds. First, we show (1.8). Take ω n to be a δ -separated set of n points in S d − with δ being as large as possible. By (2.3),(2.7) n − d − < δ < n − d − when n is sufficiently large. Note that the spherical caps of radius δ/ ω n are pairwise non-overlapping.Let now u ∈ S d − be the unit direction vector of the largest k -term signed subset sum of u , . . . , u n , that is, the largest k -term sum of ± ω n . Then the norm of this maximal sum equals tothe sum of the k largest inner products of the vectors of ± ω n taken with u .Our goal is to find a radius R so that the cap C ( u, R ) may contain at most γk points of ± ω n with a parameter γ ∈ (0 , c ( d, n, k ) γk + (1 − γ ) k cos R. Note that the open spherical caps of radius δ/ ± ω n ∩ C ( u, R ) are allcontained in C ( u, R + δ ). On the other hand, any point of C ( u, R + δ ) may be covered by at mosttwo of the interiors of these caps. Thus, we deduce C ( u, R ) ∩ ± ω n ) γk , and therefore that(2.8) holds, if R and γ satisfy(2.9) σ (cid:16) C R + δ/ (cid:17) < γ k · σ ( C δ/ ) . This is what we will show.We are going to divide the argument again into two parts according to the magnitude of k , asdifferent parameters will be needed depending on the range.Let us first assume that 3 k < · d − . Define R to be(2.10) R = δ (cid:18) k (cid:19) d − − . Notice that R > k > 3. By (2.7) then(2.11) R = Θ (cid:16)(cid:16) kn (cid:17) d − (cid:17) . In particular, R → n → ∞ . Since δ → α > (cid:16) R + δ (cid:17) (1 + α ) (cid:16) R + δ (cid:17) and sin( δ/ > δ α )for large enough n . Thus, by (2.1), σ ( C R + δ/ ) σ ( C δ/ ) tan d − ( R + δ/ d − ( δ/ (1 + α ) d − (cid:16) R δ + 1 (cid:17) d − = (1 + α ) d − · k k · , with α = ( ) / (2( d − − 1, for sufficiently large n . This shows that (2.9) holds with R = R and γ = . Since R = δ k d − (cid:16) (cid:17) d − − (cid:16) k (cid:17) d − ! , we obtain that R > δ k d − (cid:18) − k (cid:19) d − (cid:18) (cid:19) − dd − > δ d · k d − or k ∈ [3 , √ n ]. Using that cos x < − x / x ∈ [0 , π ], by (2.8) we deduce that c ( d, n, k ) k k − R ! < k − · d k d +1 d − n − d − if n is large enough. This establishes the first estimate of (1.8) with α = · .Next, we assume that 6 · d − k n . Define R so that(2.13) tan (cid:18) R + δ (cid:19) = 12 (cid:18) k n (cid:19) d − . Note that for sufficiently large n , R > δ , since by (2.7),tan 3 δ < · δ < n − d − (cid:18) k n (cid:19) / ( d − . Let α = 2 d − − 1. Then as above, for sufficiently large values of n , we obtain using (2.1) and (2.7)that σ ( C R + δ/ ) σ ( C δ/ ) tan d − ( R + δ/ d − ( δ/ − ( d − k n · (1 + α ) d − (cid:18) δ (cid:19) − ( d − < k . Therefore, (2.9) holds with R = R and γ = . Note that (2.13) shows that tan( R + δ ) , andhence R > (cid:18) R + δ (cid:19) > · 34 tan (cid:18) R + δ (cid:19) = 14 (cid:18) k n (cid:19) d − for large enough n . Thus, (2.8) implies (using that for small enough x , cos x < − x holds) that c ( d, n, k ) k + 13 k cos R k − k · (cid:18) k n (cid:19) d − k − · k d +1 d − n − d − , which is the second estimate of (1.8) with α = · .Finally, we establish (1.9). Accordingly, assume that k > √ d e − d · n . Take ω n as before. ByTheorem 6.1.6 of [14], ω n is uniformly distributed, i.e. for every closed set D ⊂ S d − with zero-measure relative boundary, lim n →∞ ω n ∩ D ) n = σ ( D )holds. Thus, ± ω n is uniformly distributed on S d − as well.Let now ε ∈ (0 , 1) be fixed, whose values we will set later. A standard compactness argumentyields that for large enough n (2.14) 11 + ε · n · σ ( C ) < ± ω n ∩ C ) < (1 + ε ) · n · σ ( C )is valid for every spherical cap C with σ ( C ) > √ d e − d . We will assume this property from nowonwards.Let again u ∈ S d − be the unit direction vector of the largest k -term sum of ± ω n . If theelements of ± ω n are ordered according to their inner products with u in decreasing order, then | u | = P ki =1 h u, v i i . Thus, if ρ denotes the spherical distance between u and v k , then(2.15) | u | X v ∈ ( ± ω n ∩ C ( u,ρ )) h u, v i . Here, the interior of C ( u, ρ ) contains strictly less than k points of ± ω n , while C ( u, ρ ) contains atleast k points of ± ω n . Because of (2.14), this shows that(2.16) 14 √ d · e − d < 11 + ε · k n σ ( C ( u, ρ )) (1 + ε ) k n . he symmetry of ± ω n also implies that σ ( C ( u, ρ )) .Now, Theorem 13.3.1 of [14] shows that the discrete probability measure with equal point massesat the elements of ± ω n converges to σ ( . ) in the weak*-topology. This implies (see e.g. Theorem1.6.5. in the same reference) that for a fixed spherical cap C ⊂ S d − ,12 n X v ∈ ( ± ω n ∩ C ) h u, v i → Z C h u, w i d σ ( w )as n → ∞ . Therefore, there exists an index N so that for every n > N and for every spherical cap C ⊂ S d − with σ ( C ) > √ d · e − d ,(2.17) 12 n X v ∈ ( ± ω n ∩ C ) h u, v i (1 + ε ) Z C h u, w i d σ ( w ) . Let now R be the radius so that(2.18) σ ( C R ) = (1 + ε ) k n . Let ϕ be defined by (1.10). Since the function f ( x ) = x e − x is monotonically decreasing on [0 , ∞ ),the condition k > √ d e − d · n ensures that ϕ < √ d . On the other hand, since k n , we also havethat ϕ > p W (1) ≈ . − ϕ d ! d − < e − ϕ which holds for every d > < ϕ < √ d , we have that | u | X v ∈ ( ± ω n ∩ C ( u,ρ )) h u, v i ε ) n Z C ( u,ρ ) h u, w i d σ ( w ) ε ) n Z C ( u,R ) h u, w i d σ ( w )= 2(1 + ε ) n · ( d − κ d − d κ d Z R t (1 − t ) d − d t ε ) n · ( d − κ d − d κ d ϕ √ d Z R (1 − t ) d − d t + Z ϕ √ d t (1 − t ) d − d t = 2(1 + ε ) n · ϕ √ d · σ ( C R ) + 2(1 + ε ) n · κ d − d κ d − ϕ d ! d − (1 + ε ) · ϕ √ d · k + 2(1 + ε ) n · √ π · √ d + 2 d · e − ϕ = ϕ √ d · k (1 + ε ) + 2(1 + ε ) 1 √ π · r d + 2 d ! ϕ √ π · √ d · k, where we set ε to be the positive solution of the quadratic equation(1 + ε ) + 2 √ π (1 + ε ) = 4 √ π . (cid:3) e note that an estimate of the same order of magnitude than (1.9) for the reduced range k > d · n may be obtained as follows. Assume that n = md , and take ω n to be m copies of anorthonormal base in R d . It is not hard to see that if a > a · m signed vectors of ω n has norm at most √ a · m . Thus, if we set k = a · m , then we readily see thatany k -term signed sum of ω n has norm at most √ a · m = r da · √ d · k which is slightly stronger than (1.9) if a > d · π ϕ . The estimate may be then extended to every k > d · n using the monotonicity property (1.4).3. Selecting two vectors We start this section by presenting two new, essentially different proofs for the estimate of c ( d, d + 1 , 2) yielded by the Welch bound. The first uses linear dependences (for several beautifulapplications of that method, see [9]). First proof of Theorem 3. By means of (1.11), it suffices to show that for any set of d + 1 unitvectors u , . . . , u d +1 ∈ S n − , there exist indices i = j ∈ [ d + 1] so that |h u i , u j i| > d . Since the number of vectors exceeds d , they must be linearly dependent: there exist reals c , . . . , c n +1 ,not all 0, so that d +1 X i =1 c i u i = 0 . Taking norm squares and using that | u i | = 1 leads to d +1 X i =1 c i = − X i 1, then we may assume w.l.o.g. that L = lin { u , . . . , u m +1 } . The inductivehypothesis applied for the vector set { u , . . . , u m +1 } implies that there exist signs ε , ε and indices1 i < i m + 1, so that | ε u i + ε u i | > r m + 1) m > r d + 1) d , proving the assertion.Hence, we may suppose that dim( L ) = d with u , . . . , u d +1 being linearly independent. Let ρ > − ρu ∈ ∂ conv( {± u , . . . , ± u d +1 } ) = ∂P, where P = conv( {± u , . . . , ± u d +1 } ), and where conv and ∂ stand for convex hull and boundary,respectively. Since P is a polytope with 0 ∈ int( P ), − ρu belongs to a facet of P , which does notcontain 0. Hence there exist signs ε , . . . , ε d +1 such that − ρu ∈ conv( { ε u , . . . , ε d +1 u d +1 } ) . In particular we have that 0 ∈ conv( { u , ε u , . . . , ε d +1 u d +1 } ) = S, thus meaning that R( S ) = 1, where R( . ) stands for the circumradius (see [16, Proposition 2.1]).Hence, if D( S ) denotes the diameter of S , then by Jung’s theorem [23] (see also [3, Lem. 3] or [17,(3)]) we have that D( S )R( S ) > r d + 1) d . Since the diameter of S is attained between two vertices of S , this means that either | u − ε i u i | > r d + 1) d or (cid:12)(cid:12) ε i u i − ε i u i (cid:12)(cid:12) > r d + 1) d , for some i ∈ { , . . . , d + 1 } or some 2 i < i d + 1, yielding the assertion.If equality holds, then we must have equality in Jung’s theorem. Therefore, the set of vertices { u , ε u , . . . , ε d +1 u d +1 } form the vertex set of a regular simplex, as desired. (cid:3) Next, we turn to estimates for large values of n . Proof of Theorem 4. We will use the fact that for two unit vectors u, v ∈ S d − of geodesic distance δ π , we have | u + v | = 2 cos δ , and therefore(3.3) 2 − δ < | u + v | < − δ . First, we show the lower bound. To this end, by (3.3), it suffices to show that if n is large enough,then for any set ω n = { u , . . . , u n } ⊂ S d − , there exist two vectors of the set ± ω n of 2 n vectors,whose geodesic distance is at most δ := 1 . · √ n − / ( d − . This is indeed guaranteed by (2.3),since if d is large enough, the maximal cardinality of a δ -separated set in S d − may not exceed23 ( d − / sin − ( d − δ · − ( d − / < . d − (cid:16) √ δ (cid:17) d − = n. Now, let us turn to the upper bound. Set δ = 0 . · n − / ( d − , and let X = { x , . . . , x m } be amaximal δ -separated subset of S d − with respect to the the Euclidean distance this time. Then X is a maximal δ ′ -separated subset of S d − , with δ < δ ′ < . δ if n is large enough. Consequently,by (2.3),(3.4) m > √ π sin − ( d − (1 . δ ) > (1 . δ ) − ( d − > . d − n. efine a graph G on the vertex set [ m ] = { , , . . . , m } as follows: two non-equal indices i and j are connected if and only if | x i + x j | < δ . We are going to bound the maximal degree in G .For every i ∈ [ m ], let B i be the d -dimensional ball of radius δ / x i . Take now anarbitrary i ∈ [ m ], and let B ′ be the ball of radius δ / − x i . Note that if ij is an edgeof G , then B ′ and B j intersect. On the other hand, B j and B j ′ do not overlap when j = j ′ . Itreadily follows by a simple geometric argument that the number of j ’s connected to i is at most τ d +1, where τ d is the kissing number in d dimensions: the maximal number of non-overlappingequal-sized spheres in R d , all touching a central sphere of the same size. Indeed, by using the factthat X is δ -separated, there exists at most one index j so that | x i + x j | < δ / 2. On the otherhand, assuming that both j and j ′ are adjacent to i , the distance between − x i and x j , resp., x j ′ isat least δ / 2, and using that | x j − x j ′ | > δ , we see that projecting x j and x j ′ radially from x i tothe sphere of radius δ centred at x i does not decrease their distance.A classical result of Kabatiansky and Levenshtein [24] states that τ ( d ) . d (1+ o (1)) . Accordingly, for large enough d , we may assume that τ d < . d − 2. Then, by the previousarguments, the maximum degree of G , denoted by ∆, is at most τ d + 1 < . d − Y be a maximal independent set in G . The cardinality of Y is the independence number of G , which is at least m/ (1 + ∆) by Brook’s bound [18, 11]. Accordingly, using (3.4), we obtainthat | Y | > m > . d − n . d > n. Thus, we may set ω n to be a set of n distinct vectors from Y . For any x i , x j ∈ ω n , we have that | x i − x j | > δ and | x i + x j | > δ . Then, using that | x i − x j | + | x i + x j | = 4,max {| x i − x j | , | x i + x j | } − δ , which implies that any signed sum of two distinct elements of ω n is bounded above by2 − δ < − . n − d − . (cid:3) We note that a slightly weaker upper bound may be obtained simply by defining edges of G corresponding to pairs i, j such that | x i + x j | < δ / 2. Then the degree of any vertex may beat most 1, which would result in an m/ G . Thisadvantage is, however, balanced out by the weaker distance bound.4. Sharp bounds in the plane First, we state and prove a simple lemma. To that end, we identify S with the complex unitcircle. For any u ∈ S of the form u = e iψ , ψ is called the angle of u . Lemma 1. Assume that ϕ ∈ [0 , π ] , and that the unit vectors u , . . . , u k ∈ S all have angles in theinterval [0 , ϕ ] . Then (4.1) | u + . . . + u k | > k cos ϕ when k is even, and (4.2) | u + . . . + u k | > r k − k + 1) cos ϕ when k is odd.Proof. Let v = e iϕ/ ∈ S . Then h u i , v i > cos ϕ olds for every i ∈ [ k ]. Consequently, D k X i =1 u i , v E > k cos ϕ . Clearly, this quantity is also a lower bound on the norm of P ki =1 u i . When k is even, we reach(4.1). That this bound is sharp is shown by considering the family consisting of k/ k/ e iϕ .Let us now assume that k is odd, and that u , . . . , u k are so that P ki =1 u i = u = e iψ has minimalnorm. Clearly, ψ ∈ [0 , ϕ ]. We may assume that the vectors u , . . . , u l are of angle at most ψ , while u l +1 , . . . , u k have angle in ( ψ, ϕ ]. It is easy to see that replacing any u i with 1 for i l , or any u j with e iϕ for j ∈ [ l + 1 , k ] results in decreasing the norm of P ki =1 u i , unless the vectors to be replacedare already 1 or e iϕ . Therefore, the extremal systems consist of copies of these two vectors at theends of the circular arc of length ϕ . If l < ( k − / 2, then swapping one copy of e iϕ by 1 againresults in decreasing the norm of the sum. Thus, by symmetry, we conclude that the minimumnorm is attained at the vector system consisting of ( k − / k + 1) / e iϕ .Applying the law of cosines leads to (4.2), which is again a sharp bound. (cid:3) Proof of Theorem 5. In light of Lemma 1, it suffices to show that from any set of 2 n unit vectorsof the form { u , − u , . . . , u n , − u n } ⊂ S , we may select k vectors which belong to an arc of anglenot larger than ( k − π/n . Order the vector system with respect to positive orientation along thecircle, and re-index the vectors as v , v , . . . , v n according to this ordering. For any i ∈ [1 , n ], let α i be the angle between v i and v i +1 , and β i be the angle between v i and v i + k − (with the indicesbeing understood modulo 2 n ). Then P ni =1 α i = 2 π , and for every i , β i = k − X j =0 α i + j . Hence, n X i =1 β i = ( k − n X i =1 α i = 2( k − π. Thus, there exists an index i for which β i ( k − π/n , which was our goal to prove.Now, let us turn to the case of equality. The above bound may only be sharp for vector systemsfor which β i = ( k − π/n for every i . On the other hand, in order for the estimate of Lemma 1being sharp, one needs that {± u , . . . , ± u n } consists of copies of a given point set with multiplicity k/ k ), or with multiplicity at least ( k − / k +1) / 2, alternatingly (for oddvalues of k ). Combining these two conditions, we deduce that the vector system {± u , . . . , ± u n } must be the vertex set of a regular ((2 n ) / ( k − S , with each of its verticesbeing taken with multiplicity k − 1. Given that {± u , . . . , ± u n } is an antipodal set, we also derivethat (2 n ) / ( k − 1) must be even, that is, k − n . When this condition holds, the vectorsystem described above indeed yields equality in (1.15) and (1.16). (cid:3) Extremality of the simplex in even dimensions Proof of Theorem 6. Let ε provide a sum of maximal norm:(5.1) u = d +1 X i =1 ε i u i . Since d is even, and multiplying all coefficients by − d +1 X i =1 ε i > . y Proposition 1, h ε i u i , u i > i . Therefore, since ε i = ± h ( ε i + 1) u i , u i > ε i holds for every i (note that (5.3) holds trivially for ε i = − P u i = 0, (5.1) leadsto u = d +1 X i =1 ( ε i + 1) u i . Therefore, summing (5.3) over all the indices i , and applying (5.2), | u | = d +1 X i =1 h ( ε i + 1) u i , u i > d + 1 + d +1 X i =1 ε i > d + 2 , which is the desired estimate. (cid:3) Acknowledgements The authors are grateful to A. Polyanski for the inspiring conversations and to D. Hardin andE. Saff for the useful advices. References [1] G. Ambrus, S. Nietert, Polarization, sign sequences and isotropic vector systems. Pacific J. Math. (2019),no. 2, 385–399.[2] G. Ambrus, I. B´ar´any and V. Grinberg, Small subset sums . Linear Algebra Appl. (2016), 66–78.[3] K. M. Ball, Ellipsoids of maximal volume in convex bodies . Geom. 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Theory (1974),397–399. Gergely Ambrus, Alfr´ed R´enyi Institute of Mathematics, 13-15. Re´altanoda u., 1053 Budapest,Hungary Email address , G. Ambrus: [email protected] Bernardo Gonz´alez Merino, Universidad de Murcia, Facultad de Educaci´on, Departamento deDid´actica de las Ciencias Matem´aticas y Sociales, 30100-Murcia, Spain Email address , B. Gonz´alez Merino: [email protected]@um.es