Additive combinatorics with a view towards computer science and cryptography: An exposition

Additiv e Com binatorics with a view to w ards Computer Science and Cryptograph y An Exp osition Kho dakhast Bibak Department of Combinatori cs and Optim izati on Universit y of W aterl o o W aterlo o, On tar io, Canada N 2L 3G1 kbib ak@u wate rloo.ca Octob er 25, 2012 Abstract Recen tly , additiv e com binatorics has blossomed in to a vibrant a r ea in mathematical sciences. But it seems to b e a diﬃcult area to deﬁn e – p erh aps b ecause of a blend of ideas and tec hniques from sev eral seemingly unr elated con texts whic h are used there. One migh t sa y that additiv e com binatorics is a branc h of mathematics concerning the study of com binatorial pr op erties of algebraic ob jects, for instance, Ab elian groups, rings, or ﬁelds. This emerging ﬁeld has s een tr emendous adv a n ces o v er the last few y ears, and h as r ecen tly b ecome a fo cus of atten tion among b oth mathematicians and computer scien tists. This fascinating area h as b een enric hed by its formidable links to com binatorics, num b er theory , h arm onic analysis, ergo dic theory , and some other branc h es; all deeply cross-fertilize eac h other, h olding grea t promise for all of them! In this exp osition, w e attempt to p ro vide an o v erview of s ome breakthroughs in this ﬁeld, together w ith a n umber of seminal app lications to sundry parts of mathematics and some other disciplines, with emphasis on compu ter science and cryptography . 1 In tro duct ion A dditive c ombinatorics is a comp elling and fast gro wing area of researc h in mathematical sciences , and the goal of this pap er is to surv ey some of the recen t dev elopmen ts and notable accomplishmen ts of the ﬁeld, focusing on b o th pure res ults and applications with a view to wards computer science a nd cryptogra ph y . Se e [ 321 ] fo r a b o ok on a dditiv e com binato rics, [ 237 , 238 ] for t w o b o o ks on additiv e n um b er theory , and [ 330 , 339 ] for t w o surv eys on additiv e 1 com binatorics. About additiv e com binatorics o v er ﬁnite ﬁelds and its applications, the reader is referred to the v ery recen t and excellen t surv ey by Shpar linski [ 291 ]. One might say that additive com binatorics studies combinatorial prop erties of algebraic ob jects, fo r example, Ab elian groups, rings, or ﬁelds, and in fact, fo cuses on the interpla y b et wee n com binatorics, num b er theory , harmonic analysis, ergo dic theory , and some other branc hes. Green [ 151 ] describ es additiv e com binat orics as t he follow ing: “additiv e com bi- natorics is the study of ap pr oximate m athematic al structur es suc h as a ppro ximate groups, rings, ﬁelds, p olynomials and homomorphisms”. Appr oximate gr oups can b e view ed as ﬁ- nite subsets o f a gr oup with the prop erty that they are almost closed under m ultiplicatio n. Appro ximate gr oups and their a pplications (f or example, to expander g raphs, group theory , probabilit y , mo del theory , and s o on) form a v ery activ e and promising area of researc h in ad- ditiv e com binator ics; the pap ers [ 64 , 65 , 67 , 68 , 151 , 190 , 315 ] con tain many de velopme n ts on this area. G o we r s [ 145 ] describes additiv e combin a torics as the follo wing: “additiv e com bina- torics fo cuses on three classes of theorems: de c o mp osition the or ems , appr oximate structur al the or em s , and tr ansfer enc e principles ”. These descriptions seem to b e mainly inspired b y new directions of this area. T ec hniques and approache s applied in additiv e com binatorics are often extremely so- phisticated, and may ha v e ro ots in sev eral unexpected ﬁelds o f mathematical sciences. F or instance, Hamidoune [ 172 ], through ideas from connectivit y pro p erties of graphs, established the so-called is op erimetric metho d , whic h is a strong to ol in additive com binatorics; see also [ 173 , 174 , 175 ] and the nice surv ey [ 278 ]. As another example, Nathanson [ 239 ] employ ed K¨ onig’s inﬁnity lemma on the existenc e o f inﬁnite paths in certain inﬁnite graphs, a nd in tro- duced a new class of additiv e bases, and also a generalization of the Erd˝ os-T ur ´ an conjecture ab out additive ba ses o f the p ositive integers. In [ 245 ] the authors emplo y ed to ols from co d- ing theory t o estimating Dav enp ort constan ts. Also, in [ 15 , 2 30 , 23 1 , 26 8 , 319 ] informatio n- theoretic tech niques are used to study sumset inequalities. V ery recen tly , Alon et al. [ 5 , 6 ], using graph-theoretic metho ds, studied sum-free sets of order m in ﬁnite Ab elian gro ups, and also, sum-free subsets of the set [1 , n ]. Add it iv e com binatorics problems in matrix rings is another active area of researc h [ 53 , 55 , 6 7 , 82 , 83 , 114 , 1 25 , 133 , 134 , 183 , 184 , 206 , 29 8 ]. A celebrated result by Szemer ´ edi, known as Szemer ´ edi’s theorem (see [ 11 , 12 , 29 , 122 , 143 , 144 , 158 , 161 , 2 36 , 246 , 255 , 256 , 2 59 , 260 , 3 04 , 310 , 311 , 329 ] f or diﬀeren t pro ofs of this theorem), states tha t ev ery subset A of the in tegers with p o sitiv e upp er densit y , that is, lim sup N →∞ | A ∩ [1 , N ] | / N > 0, has arbitrary long arithmetic progressions. A stunning breakthrough of Green and T ao [ 154 ] (that answ ers a long-standing and fo lkloric conjecture b y Erd˝ os on arithmetic pro gressions, in a sp ecial case: the primes) says that primes con ta in arbitrary long arithmetic progressions. The fusion of metho ds and ideas f rom com binatorics, n um b er theory , harmo nic analysis, and ergo dic theory used in its pro o f is v ery impressiv e. Additiv e com binatorics has recen tly found a great deal of r emark able applications to computer science and cryptograph y; for example, to expanders [ 20 , 21 , 38 , 44 , 52 , 53 , 54 , 55 , 62 , 63 , 66 , 1 05 , 1 06 , 167 , 206 , 283 , 34 2 ], extractors [ 19 , 20 , 26 , 28 , 38 , 41 , 103 , 104 , 10 7 , 167 , 182 , 217 , 346 , 3 50 ], pseudorandomness [ 33 , 223 , 226 , 227 , 30 1 ] (also, [ 331 , 334 ] a re tw o surv eys and [ 335 ] is a monograph on pseudorandomness), prop ert y testing [ 31 , 176 , 177 , 18 1 , 203 , 204 , 270 , 281 , 332 ] (see also [ 137 ]), complexit y theory [ 27 , 28 , 30 , 3 9 , 244 , 350 ], hardness ampliﬁcation [ 301 , 338 , 340 ], probabilistic c hec k able pro ofs (PCPs) [ 27 1 ], information theory 2 [ 15 , 230 , 231 , 268 , 319 ], discrete logarithm based rang e proto cols [ 77 ], no n-in teractive zero- kno wledge (NIZK) pro ofs [ 220 ], compression functions [ 192 ], hidden shifted p o we r problem [ 57 ], and Diﬃe-Hellman distributions [ 36 , 37 , 7 5 , 1 21 ]. Additiv e combinatorics also has imp ortant applications in e-v oting [ 77 , 22 0 ]. Recen tly , Bourgain et al. [ 51 ] ga ve a new explicit construction of matrices satisfying the R estricte d Isometry Pr op erty (RIP) using ideas from additive combinatorics. RIP is related to the matrices whose b ehav ior is nearly orthonormal (at least when acting on suﬃcien tly sparse v ectors); it has sev eral applications, in particular, in compressed sensing [ 71 , 72 , 73 ]. Metho ds from additiv e com binatorics provid e strong tec hniques for studying the s o -called thr e shold p h enomena , wh ic h is itself of signiﬁcan t imp orta nce in com binator ics, computer sci- ence, discrete probability , statistical ph ysics, and economics [ 2 , 34 , 60 , 119 , 120 , 198 ]. T here are also v ery strong connections b etw een ideas of a dditiv e combinatorics and the theory of r andom matric es (see, e.g., [ 322 , 323 , 341 ] and the references therein); the latter themselv es ha ve sev eral a pplications in man y areas of n umber theory , combinatorics, computer science, mathematical and theoretical ph ysics, c hemistry , and so on [ 1 , 69 , 74 , 141 , 241 , 265 , 323 , 3 24 ]. This area also has man y applications to group theory , analysis, expo nen tial sums, expanders, complexit y t heory , discrete geometry , dynamical systems, and v arious o ther scien tiﬁc disci- plines. Additiv e com binatorics has seen v ery fast adv ancemen ts in t he wak e of extremely deep w ork o n Szemer ´ edi’s theorem, the pro of of the existence of long APs in the primes by G reen and T ao, and generalizations and applications o f the sum-pro duct problem, and con tinues to see signiﬁcant progress (see [ 96 ] for a collection of op en problems in this area). In the next se ctio n, w e review Szem er´ edi’s and Green-T ao theorems (and their generalizations), t wo cornerstone breakthroughs in additiv e com binatorics. In the third section, w e will deal with the sum-pro duct pro blem: ye t another la ndmark ac hiev emen t in additiv e com bina- torics, and consider its generalizations and applications, esp ecially to computer science a nd cryptograph y . 2 Szemer ´ edi’s and Gree n-T ao Theorems, and Their Gen- eralizations R amsey the ory is concerned with the phenomenon that if a suﬃcien tly large structure (com- plete gr aphs, ar ithmetic prog ressions, ﬂat v arieties in vector spaces, etc.) is partitioned arbitrarily into ﬁnitely man y substructures, then at least one substructure has necess a rily a particular prop ert y , and so total disorder is imp ossible. In fact, Ramsey theory seeks general conditions to guarantee the existence of substructures with regular prop erties. This theory has man y applications, for example , in num b er theory , algebra, geometry , top ology , functional ana lysis (in particular, in Banach space theory), set theory , logic, ergo dic the- ory , informa tion theory , and theoretical computer science (see, e.g., [ 2 63 ] a nd t he references therein). R amsey’s the or em sa ys that in any edge coloring of a suﬃcien tly larg e complete graph, one can ﬁnd mono chromatic complete subgraphs. A nice result of the same spirit is van der Waer den ’s the or em : F or a give n k and r , there exists a num b er N = N ( k , r ) suc h that if the in tegers in [1 , N ] ar e colored using r colors, then there is a non trivial mono c hro- 3 matic k -t erm arithmetic progression ( k - AP). In t uitiv ely , this theorem asserts that in any ﬁnite coloring of a structure, one will ﬁnd a substructure of the same type a t least in one of the color classes. Note that the ﬁnitary and inﬁnitary v ersions of the v an der W aerden’s theorem are equiv alen t, through a compactness argumen t. One landmark result in Ramsey theory is the Hales-Jew ett theorem [ 171 ], whic h w as ini- tially in tro duced as a t o ol for analyzing certain kinds of games. Before stat ing this t heorem, w e need to deﬁne the concept of com bina torial line. A c ombinatorial line is a k - subset in the n - dimensional grid [1 , k ] n yielded from some template in ([1 , k ] ∪ {∗} ) n b y replacing the sym b ol ∗ with 1 , . . . , k in turn. The Hales-Jew ett theorem states that for ev ery r and k there exists n suc h that ev ery r -coloring of the n - dimensional grid [1 , k ] n con tains a com binatoria l line. Roughly sp eaking, it sa ys that for ev ery mu lt idimensional grid whose faces a re colo red with a num ber of colors, there m ust necessarily b e a line of faces of all the same color, if the dimension is suﬃcien tly large (dep ending on the nu mber of sides of the gr id and the n umber of colors). Note that instead of seeking arithmetic progressions, the Hales-Jew ett theorem seeks combinatorial lines. This theorem has ma n y in t eresting consequence s in Ramsey the- ory , t wo of whic h, are v an der W aerden’s theorem and its m ultidimensional v ersion, i.e., the Gallai-Witt theorem ( see, e.g., [ 132 , 149 , 19 7 ] for further information). Erd˝ os and T ur´ an [ 11 3 ] prop osed a v ery strong form of v an der W aerden’s theorem – the densit y v ersion of v an der W aerden’s theorem. They conjectured that arbitrarily long APs app ear not only in ﬁnite partitions but also in e very s uﬃcien tly dense subse t of p ositiv e in tegers. Mor e precisely , the Erd˝ os-T ur´ an conjecture states that if δ and k are give n, then there is a num b er N = N ( k , δ ) such that an y set A ⊆ [1 , N ] with | A | ≥ δ N con tains a non-trivial k - AP . Ro th [ 264 ] emplo y ed metho ds from F ourier analysis (or more sp eciﬁcally , the Har dy-Littlewo o d cir cle metho d ) to prov e the k = 3 case of the Erd˝ os-T ur´ an conjecture (see also [ 32 , 35 , 43 , 98 , 110 , 165 , 213 , 234 , 242 , 272 , 274 , 2 87 , 296 ]). Szemer ´ edi [ 303 ] v eriﬁed the Erd˝ os-T ur´ an conjecture f or arithmetic progress ions of length four. Finally , Sze mer´ edi [ 304 ] b y a tour de force o f sophisticated combin atorial argumen ts pro v ed the conjecture, no w kno wn as Szemer´ edi’s the or e m – one of the milestones of com binato rics. Roughly sp eaking, this theorem states that long arit hmetic progressions are v ery widespread and in f act it is not p ossible to completely get rid o f them fro m a set of p ositiv e integers unless w e can con tract the set (suﬃc iently) to mak e it of dens ity zero. Laba a nd Pramanik [ 211 ] (also see [ 247 ]) pro ved that ev ery compact set of reals with Lebesgue measu re z ero suppor ting a probabilistic measure satisfying appropriate dimensionality a nd F ourier deca y conditions m ust con tain non-trivial 3APs. Conlon and Gow ers [ 94 ] considered Szemer ´ edi’s theorem, a nd a lso sev eral o ther com bi- natorial theorems suc h as T ur´ an’s theorem a nd Ramsey’s theorem in sparse random s ets. Also, Szemer ´ edi-ty p e problems in v ario us structures other than in tegers ha v e b een a fo cus of signiﬁcant amount of w ork. F or instance , [ 150 , 2 19 ] consider these kinds of problems in the ﬁnite ﬁeld setting. V ery recen tly , Bateman and K atz [ 22 ] (also see [ 23 ]) ac hiev ed new b ounds for the c ap se t pr oblem , which is basically Roth’s pro blem, but in a v ector space ov er ﬁnite ﬁelds (a set A ⊂ F N 3 is called a c ap set if it con tains no lines). A salien t ing redien t in Szemer ´ edi’s pro of (in additio n to v an der W aerden’s theorem) is the Szemer´ edi r e gularity lemma . This lemma was conceiv ed sp eciﬁcally for the purp ose of this pro of, but is now, by itself, one of the most p o werful to ols in extremal graph theory 4 (see, e.g., [ 205 , 258 ], whic h a re t w o surv eys on this lemma and its applications). Ro ughly sp eaking, it asserts that the v ertex set o f ev ery (large) graph can b e partitioned in t o relativ ely few parts suc h that the subgraphs b et wee n the parts are random-like . Indeed, this result states that each la rge dense gra ph ma y be decomp osed in to a low-complexit y part and a pseudorandom part (note that Szemer ´ edi’s regularit y lemma is the arch etypal example of the dich o tomy b etwe en structur e and r andomness [ 312 ]). The lemma has found numerous applications not only in gra ph theory , but also in discrete geometry , additiv e combinatorics, and computer science. F or example, as T revisan [ 330 ] mentions, to solv e a computatio nal problem o n a given graph, it might b e easier to ﬁrst construct a Szemer ´ edi appro ximation – this resulted approximating graph has a simpler conﬁguration and would b e easier to treat. Note t hat the signiﬁcance of Szemer ´ edi’s regularity lemma go es b ey ond graph theory: it can b e reformulated as a result in information theory , appro ximation theory , as a compactness result on the completion of the s pa ce of ﬁnite gra phs, etc. (see [ 222 ] and the references therein). V ery recen t ly , T ao and Green [ 158 ] established an arithmetic r e gularity lemma and a complemen t ary arithmetic c ounting lemma that hav e sev eral applications, in pa rticular, an astonishing pro of of Szemer ´ edi’s theorem. The triangle r emoval lemma established by Ruzsa and Szemer ´ edi [ 269 ] is one of the mo st notable applications of Szemer ´ edi’s regularity lemma. It asserts tha t each gra ph of order n with o ( n 3 ) triang les can b e made triangle-free by remo ving o ( n 2 ) edges. In ot her w ords, if a graph has asymptotically few triangles then it is asymptotically close to b eing triangle-free. As a clev er application of this lemma, Ruzsa and Szemer ´ edi [ 269 ] o btained a new pro of o f Roth’s theorem (see also [ 296 ], in whic h the author using the triangle remo v al lemma prov es Roth type theorems in ﬁnite gro ups). Note that a generalization of the triangle remo v al lemma, kno wn as sim plex r emoval lemma , can b e used to deduce Szem er ´ edi’s theorem (see [ 144 , 259 , 260 , 311 ]). T he triangle remov al lemma was extended b y Erd˝ os, F rankl, and R¨ odl [ 111 ] to the gr aph r emoval lemma , which roughly sp eaking, asserts that if a giv en graph do es not contain to o man y subgraphs of a giv en type, then all the subgraphs of this t yp e can b e remo v ed b y deleting a few edges. M o re precisely , giv en a ﬁxed graph H of order k , an y graph of order n with o ( n k ) copies of H can be made H -free b y remo ving o ( n 2 ) edges. F ox [ 115 ] ga ve a proof of the graph remo v al lemma whic h a v o ids applying Szemer ´ edi’s regularity lemma and give s a b etter b ound (also see [ 92 ]). The graph remov al lemma has man y applications in graph theory , additiv e com binatorics, discrete geometry , and theoretical computer science. One surprising application of this lemma is to the area of pr o p erty testing , which is no w a v ery dynamic area in computer science [ 7 , 8 , 9 , 10 , 13 , 137 , 138 , 18 7 , 254 , 257 , 261 , 262 , 270 ]. Prop ert y testing t ypically refers to the existence of sub-linear time probabilistic algorithms (called testers), whic h distinguish b etw een ob jects G (e.g., a graph) ha ving a given prop ert y P (e.g., bipartiteness) and those b eing far aw a y (in an appropriate metric) from P . Prop erty testing algorithms ha v e b een recen tly designed and utilized for man y kinds of ob jects and prop erties, in particular, discrete properties (e.g., graph prop erties, discrete functions, and sets of intege rs), geometric prop erties, alg ebraic prop erties, etc. There is a (growing) n um b er of pro o fs of Szemer ´ edi’s theorem, arguably sev en teen pro ofs to this date. One suc h elegant proo f that uses ideas from mo del theory w as giv en b y T ow sner [ 329 ]. F or another mo del theory based p ersp ective , see [ 232 ], in which the authors giv e stronger regularity lemmas for some classes of g raphs. 5 In fact, one might claim that man y of these pro ofs hav e t hemselv es o p ened up a new ﬁeld of researc h. F ursten b erg [ 122 ] b y rephrasing it as a problem in dynamic al systems , and then applying sev eral p ow erful tec hniques from ergo dic t heory ac hieve d a nice pro of of the Szemer ´ edi’s theorem. In fact, F ursten b erg presen ted a corr espo ndence b et w een problems in the subsets of p ositiv e densit y in the in tegers and recurrence problems for sets of p ositiv e measure in a pr ob abili ty me asur e pr eserving system . This observ ation is now kno wn as the F urstenb er g c orr esp ondenc e principle . Er go dic the ory is concerne d with the long- term b eha vior in dynamical system s from a statistical p oint of view (see, e.g., [ 108 ]). This area and its formidable w ay of thinking ha v e made man y strong connections with sev eral branc hes of mathematics, including com binatorics, n umber theory , co ding theory , group theory , and harmonic analysis; see, for example, [ 207 , 2 08 , 2 09 , 2 10 ] and the references therein for some connections b etw een ergo dic theory and additive com binatorics. This ergo dic-theoretic metho d is one of the most ﬂex ible know n pro o fs, and h a s been v ery success ful at reaching considerable g eneralizations of Szemer ´ edi’s theorem. F ursten b erg and Katznelson [ 123 ] obtained the multidimensio nal Szemer´ edi the or em . Their proof relies on the concept of multiple r e curr enc e , a p ow erful to ol in the in t eraction b et we en ergo dic theory and additiv e com binatorics. A purely combinatorial pro of of t his theorem w as obtained ro ughly in par allel by Gow ers [ 144 ], and Nagle et a l. [ 236 , 255 , 2 56 , 25 9 , 260 ], and subsequen tly by T ao [ 311 ], via establishing a hyp er gr aph r emoval lemma (see a lso [ 257 , 31 4 ]). Also, Austin [ 11 ] prov ed the theorem v ia b oth e r go dic-theoretic and com binatorial approac hes. The m ulti- dimensional Szemer ´ edi theorem w as signiﬁcan tly generalized by F ursten b erg and Katznelson [ 124 ] (via ergo dic-theoretic approaches ) , and Austin [ 12 ] (via b ot h ergo dic-theoretic and com binatorial approache s), to the dens i ty Hales-Jewett the or em . The densit y Hales-Jew ett theorem states tha t for eve ry δ > 0 there is some N 0 ≥ 1 suc h that whenev er A ⊆ [1 , k ] N with N ≥ N 0 and | A | ≥ δ k N , A con ta ins a com binatorial line. Recen tly , in a massiv ely collab orative online pro ject, namely Polyma th 1 (a pro ject that originated in G o we rs’ blog), the Polymath team found a purely com binatorial pro o f of the densit y Hales-Jew ett theorem, whic h is also the ﬁrst o ne pro viding explicit b o unds for how large n needs to b e [ 246 ] (also see [ 240 ]). Suc h b ounds could not b e o btained through the ergo dic-theoretic metho ds, since these pro ofs rely on the Axiom of Choice. It is w orth men tioning that this pro ject w as selected as one of t he TIME Magazine’s Best Ideas of 2009. F urstenberg’s pro of gav e rise to the ﬁeld of er go dic R amsey the ory , in which arithmetical, com binatorial, and geometrical conﬁgurations pr eserv ed in (suﬃcien tly large) substructures of a structure, are treated via ideas and t ec hniques from ergo dic theory (or more sp eciﬁcally , m ultiple recurrence). Ergo dic Ramsey theory has since pro duced a high nu m b er of com bi- natorial results, some of whic h ha v e y et to b e obtained b y other means, and has also giv en a deeper understanding of the structure of measure preserving system s. In fact, ergo dic theory has b een used to solve problems in Ramsey theory , and recipro cally , Ramsey theory has led to the disco v ery of new phenomena in ergo dic theory . Ho w ev er, t he ergo dic-theoretic metho ds and the inﬁnitary nature of their tec hniques ha v e some limitations. F or example, these methods do not pro vide an y eﬀectiv e b ound, since, as w e already men tioned, they rely on the Axiom of Choice. Also, despite v an der W aerden’s t heorem is not directly used in F ursten b erg’s pro of, probably any eﬀort to mak e the pro of quan titativ e w ould result in rapidly growing functions . F urthermore, the ergo dic-theoretic metho ds, to this da y , hav e the 6 limitation of o nly b eing able to deal with sets of p ositiv e densit y in the in tegers, although this densit y is allo w ed to b e arbitrarily small. Ho w ev er, Green and T ao [ 15 4 ] discov ered a tr ans- fer enc e principle whic h a llo we d one to reduce problems on structures in sp ecial sets of zero densit y (suc h as the primes) to problems on sets of positiv e densit y in the in tegers. It is w or th men tioning that Conlon, F ox, and Zhao [ 93 ] established a tr a n sfer enc e principle extending sev eral classical extremal graph theoretic results, including the remo v a l lemmas for graphs and groups (the latter leads t o an extension of Roth’s theorem), the Erd˝ os-Stone-Simonovits theorem and Ramsey’s theorem, to sparse pseudorandom graphs. Go wers [ 143 ] generalized the a rgumen ts previously studied in [ 142 , 264 ], in a substan tial w ay . In fact, he emplo y ed com binatorics, generalized F ourier analys is, and in v erse arith- metic combinatorics (including mu lt ilinear ve rsions of F r eiman ’s the or em on sumsets, and the Ba l o g-Sze mer´ edi the or em ) to reprov e Szemer ´ edi’s theorem with explicit b ounds. Note that F ourier analysis has a wide range of a pplications, in particular, to cryptography , hard- ness of appro ximatio n, signal pro cessing, threshold phenomena for probabilistic mo dels suc h as random graphs and p ercolations, a nd many other disciplines. Gow ers’ article in tro duced a kind of higher de gr e e F ourier analysis , whic h has b een f urther dev elop ed b y Green and T ao. Indee d, G o wers initiated the study of a new measure of functions, now referred to as Gowers (unifo rm ity) n orms , that resulted in a b etter understanding of the no tion of pseudor andomness . The Gow ers norm, whic h is an imp ortan t sp ecial case of noise correlation ( in tuitive ly , the noise c orr ela tion b et ween t wo functions f and g measures ho w muc h f ( x ) and g ( y ) correlate o n random inputs x and y whic h are correlated), enjo ys man y prop erties and applications, and is no w a v ery dynamic area of researc h in mathematical science s; see [ 14 , 153 , 157 , 159 , 163 , 1 64 , 17 6 , 181 , 2 03 , 22 1 , 224 , 225 , 233 , 270 , 27 1 ] for mor e prop erties and applications of the G o we r s norm. Also, the b est kno wn b ounds for Szemer ´ edi’s theorem are obtained throug h the so-called inverse the or ems for Go wers nor ms. Rece n tly , Green and T ao [ 161 ] (see also [ 320 ]), using the density - incremen t strategy of Roth [ 264 ] and Gow ers [ 142 , 143 ], deriv ed Szemer ´ edi’s theorem from the inverse c onje ctur es GI( s ) for the Gow ers norms, whic h w ere recen tly established in [ 164 ]. T o the b est of my know ledge, there are tw o types of in vers e theorems in a dditiv e com bi- natorics, namely the invers e sumset the or ems of F r eiman typ e (see, e.g., [ 7 8 , 117 , 118 , 152 , 155 , 275 , 299 , 306 , 307 , 308 , 318 ] and [ 116 , 238 ]), and inverse the or ems for the Gowers norms (see, e.g., [ 143 , 146 , 147 , 14 8 , 15 3 , 162 , 156 , 159 , 163 , 164 , 1 76 , 18 1 , 189 , 203 , 224 , 225 , 233 , 270 , 326 , 332 ]). It is interes t ing that the inv erse conjecture leads to a ﬁnite ﬁeld v ersion of Szemer ´ edi’s theorem [ 320 ]: Let F p b e a ﬁnite ﬁeld. Supp ose that δ > 0, and A ⊂ F n p with | A | ≥ δ | F n p | . If n is suﬃcien tly large dep ending on p a nd δ , then A contains an (aﬃne) line { x, x + r , . . . , x + ( p − 1) r } for some x, r ∈ F n p with r 6 = 0 (actually , A con ta ins an aﬃne k -dimensional subspace, k ≥ 1). Supp ose r k ( N ) is the cardinality of the largest subset of [1 , N ] containing no nontrivial k -APs. Giving asymptotic estimates on r k ( N ) is an import an t in v erse problem in additiv e com binatorics. Behrend [ 24 ] pro ved that r 3 ( N ) = Ω N 2 2 √ 2 √ log 2 N . log 1 / 4 N ! . 7 Rankin [ 249 ] g eneralized Behrend’s construction to lo nger APs. Roth prov ed that r 3 ( N ) = o ( N ). In fact, he pro v ed the ﬁrst nontrivial upp er b ound r 3 ( N ) = O N log log N ! . Bourgain [ 35 , 43 ] improv ed Roth’s b ound. In fact, Bourgain [ 43 ] ga ve the upp er b ound r 3 ( N ) = O N (log log N ) 2 log 2 / 3 N ! . Sanders [ 274 ] prov ed the following upp er b o und which is the state- of-the-art: r 3 ( N ) = O N (log log N ) 5 log N ! . Blo om [ 32 ] throug h the nice techniq ue “translation of a pro of in F q [ t ] to one in Z / N Z ”, extends Sanders ’ pro of to 4 and 5 v ariables. As Blo om men tions in his pap er, man y problems of additive com binatorics migh t b e easier to attack via the approac h “t ranslating fro m F N p to F q [ t ] and hence to Z / N Z ” . Elkin [ 110 ] mana ged to impro v e Behrend’s 6 2-y ear o ld low er b ound b y a factor of Θ(log 1 / 2 N ). Actually , Elkin sho w ed that r 3 ( N ) = Ω N 2 2 √ 2 √ log 2 N . log 1 / 4 N ! . See also [ 165 ] for a short pro of of Elkin’s res ult, and [ 242 ] for constructiv e low er b ounds for r k ( N ). Sch o en and Shkredov [ 27 7 ] using ideas from the pap er of Sanders [ 273 ] and also the new pro babilistic tec hnique established by Cro ot and Sisask [ 98 ], obtained Behrend-type b ounds for linear equations in volv ing 6 or more v ar iables. Thanks to this result, one ma y see that p erhaps the Behrend-t yp e constructions are not to o far from b eing b est-p ossible. Almost all the kno wn pro ofs o f Szemer ´ edi’s theorem are based on a dichotomy b etwe en structur e and r andomness [ 312 , 316 ], whic h allo ws man y mat hematical ob jects to b e split in to a ‘structured part’ (or ‘lo w-complexit y part’) and a ‘random part’ (or ‘discorrelated part’). T ao [ 310 ] b est describes almost a ll kno wn pro o fs of Szemer ´ edi’s theorem collectiv ely as the follo wing: “Start with the set A (or some other ob ject which is a proxy for A , e.g., a graph, a h yp ergr aph, or a measure-preservin g system). F or the ob ject under consideration, deﬁne some conc ept of randomness (e.g., ε -regularit y , uniformity , small F ourier coeﬃcien t s, or w eak mixing), a nd some concept of structure (e.g., a nested sequence of arithmetically structured sets such a s prog ressions or Bohr sets, or a part ition of a v ertex set into a con trolled num b er of pieces, a collection of lar ge F o urier co eﬃcien ts, a sequence o f almost p erio dic f unctions, a to we r of compact extens ions of the trivial factors). Obtain some sort of structure theorem that splits the ob ject in to a structured comp onen t, plus a n erro r which is random relativ e to that structured comp onent. T o pro ve Szemer ´ edi’s t heorem (or a v ar ian t thereof ), o ne then 8 needs to obtain some sort of gener al i z e d von Neumann the or em [ 154 ] to eliminate the random error, and then some sort of structur e d r e curr enc e the or em for the structured comp onen t”. Erd˝ os’s famous conjecture o n APs states that a set A = { a 1 , a 2 , . . . , a n , . . . } of p ositiv e in tegers, where a i < a i +1 for all i , with the div ergen t sum P n ∈ Z + 1 a n , contains a rbitrarily long APs. If true, the theorem includes b o th Szemer ´ edi’s and Green-T ao theorems a s sp ecial cases. This conjecture seems to b e to o strong to hold, and in fact, might b e ve ry diﬃcult to attac k – it is not ev en kno wn whether suc h a set must contain a 3-AP! So, let us men tion an equiv alen t statemen t for Erd˝ os’s conjecture that ma y b e helpful. Let N b e a positive in teger. F o r a po sitiv e in t eger k , deﬁne a k ( N ) := r k ( N ) / N (note that Szem er´ edi’s theorem asserts that lim N →∞ a k ( N ) = 0, fo r all k ). It can b e prov ed (see [ 286 ]) that Erd˝ o s’s conjecture is true if and only if the series P ∞ i =1 a k 4 i con v erges for an y in teger k ≥ 3. So, to prov e Erd˝ os’s conjecture, it suﬃc es t o obtain the estimate a k ( N ) ≪ 1 / (log N ) 1+ ε , for any k ≥ 3 and for some ε > 0. Szemer ´ edi’s theorem pla ys an impor tan t role in the pro of of the Green-T ao theorem [ 154 ]: The primes con tain arithmetic progressions of a rbitrarily large length (note that the same result is v alid for eve r y subset of the primes with p ositive relativ e upper densit y). G reen and T ao [ 160 ] also prov ed t hat there is a k -AP o f primes all of whose terms are b ounded b y 2 2 2 2 2 2 2 2 (100 k ) , whic h sho ws that how far out in the primes one mus t go to warran t ﬁnding a k -AP . A conjecture (see [ 207 ]) asserts that there is a k - AP in the primes all of whose terms a re b ounded by k ! + 1. There are three fundamental ingredien ts in the pro of of the Green-T ao theorem (in fact, there are man y similarities b et we en G reen and T ao’s approac h and the ergo dic-theoretic metho d, see [ 188 ]). The ﬁrst is Szemer ´ edi’s theorem itself. Since the primes do not ha ve p ositiv e upp er densit y , Szemer ´ edi’s theorem cannot b e dir ectly applied. The second ma jor ingredien t in the pro of is a certain tr ansfer enc e principle that allows one to use Szemer ´ edi’s theorem in a more general setting (a generalization of Szemer ´ edi’s theorem to the pseu- dor andom sets , whic h can hav e zero densit y). The last ma jor ingr edien t is applying some notable features of the primes and the ir distribution through results of Goldston and Yildirim [ 139 , 140 ], and pro ving the fact that t his generalized Szeme r´ edi theorem can b e eﬃcien tly applied to the primes, and indeed, the set o f primes w ill ha ve the des ir ed pseud o random prop erties. In fact, G reen and T ao’s pro of emplo ys the tec hniques applied in sev eral known pro ofs of Szemer ´ edi’s theorem and exploits a dichotom y b etw een structure a nd randomness. This pro of is based on ideas and results from sev eral branc hes of mathematics, for example, com binatorics, analytical n umber theory , pseudorandomness, harmonic analysis, and ergo dic theory . Reingold et al. [ 252 ], and G o wers [ 145 ], indep enden tly obtained a short pro of for a fundamen tal ingredien t of this pro of. T ao and Ziegler [ 325 ] (see also [ 252 ]) , v ia a transference principle for p olynomial con- ﬁgurations, extended the Green-T ao theorem to cov er p olynomial pro gressions: Let A ⊂ P b e a set of primes of p ositiv e relativ e upp er densit y in the primes, i.e., lim sup N →∞ | A ∩ 9 [1 , N ] | / |P ∩ [1 , N ] | > 0. Then, g iv en an y in t eger-v alued p olynomials P 1 , . . . , P k in one un- kno wn m with v anishing constan t terms, the set A con ta ins inﬁnitely many progressions of the form x + P 1 ( m ) , . . . , x + P k ( m ) with m > 0 (note that the sp ecial case when the p oly- nomials a re m, 2 m, . . . , km implies the previous result that there are k -APs of primes). T ao [ 313 ] prov ed the analogue in the Gaussian in tegers. Green and T ao (in view of the parallelism b et wee n the in tegers and the p olynomials ov er a ﬁnite ﬁeld) though t that the a nalogue of their theorem should b e held in t he setting of function ﬁelds; a result that w as prov ed b y L ˆ e [ 212 ]: Let F q b e a ﬁnite ﬁeld ov er q elemen ts. Then f or any k > 0, one can ﬁnd p olynomials f , g ∈ F q [ t ] , g 6≡ 0 suc h that the p olynomials f + P g are all irreducible, where P runs ov er all p olynomials P ∈ F q [ t ] of degree less than k . Moreov er, suc h structures can b e fo und in ev ery set of p ositiv e relativ e upp er densit y among the ir reducible p o lynomials. The pro of of this in teresting theorem f ollo ws the ideas of the pro o f of the Green-T ao theorem v ery closely . 3 Sum-Pro duct Problem: Its Generaliz ati ons and Ap- plications The sum-pr o duct pr oblem and its generalizations constitute a nother vibran t area in additiv e com binatorics, and ha v e led to man y seminal applications to n umber theory , Ramsey theory , computer science, and cryptograph y . Let’s start with t he deﬁnition of sumset, pro duct set, and some preliminaries. W e will follo w closely the pre sentation of T ao [ 316 ]. Let A b e a ﬁnite nonempt y set of elemen ts of a ring R . W e deﬁne the sumset A + A = { a + b : a, b ∈ A } , and the pr o duct set A · A = { a · b : a, b ∈ A } . Supp ose that no a ∈ A is a zero divisor (otherwise, A · A may b ecome v ery small, whic h lead to degene r ate cases). Then one can easily show that A + A and A · A will b e at least as large as A . The set A ma y b e almost closed under addition, whic h, for ex ample, o ccurs whe n A is an arithmetic progression or an additiv e su bgroup in the ring R (e.g., if A ⊂ R is an AP , then | A + A | = 2 | A | − 1, and | A · A | ≥ c | A | 2 − ε ), or it may b e almost closed und er multiplic a tion, whic h, for example, o ccurs when A is a geometric progression or a m ultiplicativ e subgroup in the ring R (e.g., if N ⊂ R is an AP and A = { 2 n : n ∈ N } , t hen | A · A | = 2 | A | − 1, and | A + A | ≈ | A | 2 ). Note that even if A is a dense subs et of an arit hmetic progr ession or additiv e subgroup (or a dens e subset of an geometric progr ession o r multiplicativ e subgroup), then A + A (or A · A , r espective ly) is still comparable in size to A . But it is diﬃcult f or A to b e almost closed under a ddition and m ultiplication simultaneously , unless it is v ery close to a subring. The sum-pr o duct phenomenon says that if a ﬁnite set A is not close to a subring, then either the sumset A + A or the pro duct set A · A m ust b e considerably la rger than A . The reader can refer to [ 276 ] and t he references therein to see some lo wer b ounds on | C − C | and | C + C | , where C is a con v ex set (a set of reals C = { c 1 , . . . , c n } is called c onvex if c i +1 − c i > c i − c i − 1 , for all i ). In the reals setting, do es there exist an A ⊂ R for whic h max {| A + A | , | A · A |} is ‘small’ ? Erd˝ os and Szemer ´ edi [ 112 ] ga ve a negativ e answ er to this question. Actually , they pro ve d the inequalit y max {| A + A | , | A · A |} ≥ c | A | 1+ ε for a small but p ositiv e ε , where A is a subset of t he reals. They also conjectured that max {| A + A | , | A · A |} ≥ c | A | 2 − δ , for any p ositiv e δ . Muc h eﬀorts hav e b een made to wards the v alue of ε . Elek es [ 109 ] observ ed that 10 the sum-pro duct problem has interesting connections to problems in incidence geometry . In particular, he applied the so-called Szemer´ e d i-T r otter the o r em and sho w ed that ε ≥ 1 / 4, if A is a ﬁnite set of real num b ers. Elek es’s result w as extended to complex num b ers in [ 292 ]. In the case of reals, the state-of-the-art is due to Solymosi [ 29 5 ]: one can tak e ε arbitrar ily close to 1 / 3. F or complex num b ers, So lymosi [ 293 ], using the Szemer ´ edi-T rotter theorem, pro ve d that one can tak e ε arbitrarily close to 3 / 11. V ery recen tly , Rudnev [ 267 ], again using the Szemer ´ edi-T ro tter theorem, obtained a b ound whic h is the state-of -the-art in the case of complex n umbers: one can tak e ε arbitrarily close t o 19 / 69. Solymosi and V u [ 298 ] pro v ed a sum-pro duct estimate for a sp ecial ﬁnite set of square matrices with complex en tr ies, where that set is w e l l-c onditione d (t hat is, its matrices are far fro m b eing singular). Note that If w e remov e t he latter condition (i.e., w ell-conditioned!) then the theorem will not b e tr ue; see [ 298 , Example 1.1 ]. W olﬀ [ 345 ] motiv ated by the ‘ﬁnite ﬁeld Kakey a conjecture’, formulated the ﬁnite ﬁeld v ersion of sum-pro duct problem. The Kak eya c onje ctur e sa ys that the Hausdorﬀ dimension of any subset of R n that contains a unit line segmen t in ev ery dir ection is equal to n ; it is op en in dimensions at least three. The ﬁnite ﬁeld Kakeya c onje ctur e asks for the smallest subset of F n q that con tains a line in each direction. This conjecture w as pro ved b y Dvir [ 102 ] using a clev er application of the so- called p olynomial metho d ; see a lso [ 101 ] for a nice surv ey on this problem and its applications especially in t he ar ea of rando mness extractors. The p olynomial metho d, which has pro ve d to b e ve ry useful in additive com binato rics, is roughly described as the following: Give n a ﬁeld F and a ﬁnite subset S ⊂ F n . Mu ltiv a riate p olynomials o v er F whic h v a nish on all p oints of S , usually get s ome com binatorial properties ab out S . (This has some similarities with what we usually do in alg ebraic geometry!) See, e.g., [ 197 , Chapter 16] for some basic facts about the p olynomial metho d, [ 102 , 168 , 169 , 170 ] for applications in a dditiv e com binatorics, and [ 1 07 , 167 , 300 ] fo r applications in computer science. Actually , the ﬁnite ﬁe ld vers io n (of sum-pro duct problem) b ecomes more diﬃcult, b ecause w e will encounter with some diﬃculties in applying the Szemer ´ edi- T rotter incidence theorem in t his setting. In fact, the cr ossing lemma , which is an impo rtan t ingredien t in t he pro of of Szemer ´ edi-T rotter theorem [ 302 ], relies on Euler’s formula (and so o n the top ology of the plane), a nd consequen tly do es not w ork in ﬁnite ﬁelds. Note that the pro of that Szemer ´ edi and T rotter presen ted for their theorem w a s somewhat complicated, using a com binatorial tec hnique kno wn as c el l de c om p osition [ 305 ]. When w or king with ﬁnite ﬁelds it is imp ortan t to consider ﬁelds whose order is prime and not the p ow er of a prime; b ecause in the latter case w e can take A to b e a subring whic h leads t o the degenerate case | A | = | A + A | = | A · A | . A stunning r esult in the case of ﬁnite ﬁeld F p , with p prime, w as prov ed b y Bourga in, Katz and T ao [ 61 ]. They pro v ed the follo wing: if A ⊂ F p , a nd p δ ≤ | A | ≤ p 1 − δ for some δ > 0, t hen there exists ε = ε ( δ ) > 0 suc h t hat max {| A + A | , | A · A | } ≥ c | A | 1+ ε . This result is n ow kno wn as the su m -pr o duct the or em f or F p . In fact, this theorem holds if A is not to o close to b e the whole ﬁeld. The condition | A | ≥ p δ in this theorem w as remov ed b y Bourgain, Glibic huk , and Kon y ag in in [ 59 ]. Also, note that the condition | A | ≤ p 1 − δ is necessary (e.g., if w e consider a set A consisting of all elemen ts of the ﬁeld except one, t hen 11 max {| A + A | , | A · A |} = | A | + 1). The idea f or the pro of o f this theorem is b y con t radiction; assume tha t | A + A | a nd | A · A | a re close to | A | and conclude tha t A is b eha ving v ery muc h lik e a subﬁeld of F p . Sum-product estimates f or ra tional functions (i.e., the results that one of A + A or f ( A ) is subs ta n tially larger than A , where f is a rational function) ha v e also b een treated (see, e.g., [ 19 , 70 ]). Also, note that problems of the kind ‘in teraction o f summation and addition’ are v ery imp ortan t in v a rious con t exts of additiv e com binatorics and hav e v ery in teresting applications (see [ 16 , 130 , 135 , 136 , 170 , 191 , 243 , 253 ]). Garaev [ 126 ] prov ed the ﬁrst quan titativ e sum-pro duct estimate for ﬁelds of prime o rder: Let A ⊂ F p suc h that 1 < | A | < p 7 / 13 log − 4 / 13 p . Th en max {| A + A | , | A · A | } ≫ | A | 15 / 14 log 2 / 7 | A | . Garaev’s result w a s extended and impro ve d b y sev eral authors. Rudnev [ 266 ] pro ved the follo wing: Let A ⊂ F ∗ p with | A | < √ p and p large. Then max {| A + A | , | A · A |} ≫ | A | 12 / 11 log 4 / 11 | A | . Li and Ro c he-Newton [ 215 ] prov ed a sum-pro duct estimate fo r subsets of a ﬁnite ﬁeld whose order is not prime: Let A ⊂ F p n with | A ∩ cG | ≤ | G | 1 / 2 for any subﬁeld G of F p n and any elemen t c ∈ F p n . Then max {| A + A | , | A · A |} ≫ | A | 12 / 11 log 5 / 11 2 | A | . See also [ 56 , 127 , 12 8 , 201 , 202 , 2 14 , 28 2 , 284 ] for other generalizations and impro v ements of Garaev’s result. As an application, Shpar linski [ 290 ] using Rudnev’s result [ 266 ], estimates the cardinality , #Γ p ( T ), of the set Γ p ( T ) = { γ ∈ F p : ord γ ≤ T and ord( γ + γ − 1 ) ≤ T } , where ord γ (m ultiplicative order of γ ) is the smallest p ositiv e in teger t with γ t = 1. T´ oth [ 328 ] g eneralized the Szemer ´ edi-T rotter theorem to complex p oints and lines in C 2 (also see [ 347 ] for a diﬀeren t pro of of this result, a nd a sharp result in the case of R 4 ). As another a pplication of the sum-pro duct theorem, Bourgain, Katz and T a o [ 61 ] (a lso see [ 38 ]) deriv ed an imp o rtan t Sz emer ´ edi-T rotter t yp e theorem in prime ﬁnite ﬁelds (but did not quan tify it): If F p is a prime ﬁeld, and P and L are p oints and lines in the p ro jectiv e plane o v er F p with cardinalit y |P | , |L| ≤ N < p α for some 0 < α < 2, then   { ( p, l ) ∈ P × L : p ∈ l }   ≤ C N 3 / 2 − ε , for some ε = ε ( α ) > 0. Note that it is not diﬃcult t o generalize this theorem from prime ﬁnite ﬁelds to ev ery ﬁnite ﬁeld that do es not con ta in a large subﬁeld. The ﬁrst quan titat iv e Szemer ´ edi-T r otter t yp e theorem in prime ﬁnite ﬁelds w as o btained b y Helfgott and Rudnev [ 185 ]. They show ed that ε ≥ 1 / 10678 , when | P | = |L| < p . (Note that this condition prev en ts P from b eing the en tire plane F 2 p .) This result w as extended to general ﬁnite ﬁelds (with a slightly w eak er exp onen t) b y Jones [ 193 ]. Als o , Jo nes [ 194 , 196 ] improv ed the result of Helfgott and Rudnev 12 [ 185 ] by replacing 1 / 10678 with 1 / 806 − o (1), and 1 / 662 − o (1), resp ectiv ely . A near-sharp generalization of the Szemer ´ edi-T rotter theorem to higher dimensional p oin ts and v arieties w as obtained in [ 297 ]. Using ideas from a dditiv e com binatorics ( in fact, com bining the tec hniques of cell de- comp osition and p olynomial metho d in a no v el wa y), Guth and Katz [ 170 ], ac hiev ed a near-optimal b ound fo r the Er d˝ os distinct d i s tan c e pr oblem in t he plane. They pro ved that a set o f N p oints in the pla ne has at least c N log N distinct distances (see also [ 131 ] f or some tec hniques and ideas related to this problem). Hart, Iosevic h, and Solymosi [ 179 ] obtained a new proo f of the sum-product theorem based on incidence theorems for h yp erb olas in ﬁnite ﬁelds whic h is ac hiev ed through some estimates on K lo osterman ex p onen tial sums. Some other results related to the incidence theorems can b e found in [ 95 , 178 ]. The sum-pro duct theorem has a plethora of deep applications to v arious areas suc h as incidence geometry [ 30 , 38 , 61 , 85 , 86 , 19 1 , 195 , 253 , 309 , 342 , 343 ], analysis [ 7 9 , 125 , 180 , 309 ], PDE [ 309 ], group theory [ 62 , 67 , 133 , 134 , 1 83 , 184 , 190 , 206 ], exp onen tia l a nd c haracter sums [ 18 , 36 , 37 , 38 , 39 , 40 , 4 2 , 45 , 4 6 , 47 , 48 , 57 , 58 , 59 , 81 , 129 ], n umber theory [ 17 , 18 , 42 , 46 , 54 , 55 , 81 , 84 , 89 , 90 , 91 , 243 ], com binatorics [ 20 , 62 , 53 , 309 ], expanders [ 38 , 5 2 , 53 , 54 , 55 , 62 , 66 , 20 6 , 283 , 34 2 ], extractors [ 19 , 20 , 3 8 , 41 , 88 , 103 , 182 , 346 ], disp ersers [ 20 ], complexit y theory [ 39 ], pseudorandomness [ 33 , 38 , 227 ], prop erty testing [ 137 , 270 ], hardness ampliﬁcation [ 338 , 340 ], probabilistic c hec k able pro ofs (PCPs) [ 271 ], and cryptography [ 3 6 , 37 , 75 , 121 ]. A sum-pro duct problem asso ciated with a g raph was initiated b y Erd˝ os and Szem er´ edi [ 112 ]. Alon et al. [ 4 ] studied the sum-pro duct theorems for sparse gra phs, and obta ined some nice results when the graph is a matc hing. Let us ask do es t here exist an y connection b et wee n the ‘sum-pro duct problem’ and ‘sp ec- tral graph theory’ ? Surprisingly , the answ er is ye s! In fact, the ﬁrst pap er that introduced and applied the sp ectral metho ds to estimate sum-pro duct problems ( and ev en more general problems) is the pap er b y V u [ 342 ] (see also [ 180 ], in whic h F ourier analytic metho ds were used to generalize the results b y V u). In his elegan t paper, V u relates t he sum-pro duct b ound to the expansion of certain graphs, a nd t hen via the relation of the sp ectrum (second eigen- v a lue) and expans ion one can deduce a rather strong bound. Vinh [ 336 ] (also see [ 337 ]), using ideas from sp ectral gra ph theory , deriv ed a Szemer ´ edi-T rotter t yp e t heorem in ﬁnite ﬁelds, and f rom there o btained a diﬀeren t pro of of Gar aev’s r esult [ 128 ] on sum-pro duct estimate for large subsets of ﬁnite ﬁelds. Also, Solymosi [ 294 ] applied tec hniques from sp ectral g raph theory and obtained estimates similar to those of Garaev [ 126 ] that already follo we d via to ols from exponential sums and F ourier analysis. One imp ortant ingredie n t in Solymosi’s metho d [ 294 ] is the w ell-kno wn Exp ander Mixing L emma (see, e.g., [ 186 ]) , whic h roughly sp eaking, states that o n g raphs with go o d expansion, the edges of t he graphs a re w ell-distributed, and in f act, t he n umber of edges b etw een an y t wo v ertex subsets is ab out what o ne w ould exp ect for a random g raph of that edge densit y . The generalizations o f the sum-pro duct problem to p o lynomials, elliptic curv es, and also the exp onen tia ted v ersions of the pro blem in ﬁnite ﬁelds w ere obtained in [ 3 , 1 9 , 70 , 97 , 130 , 285 , 288 , 289 , 342 ]. Also, the problem in the comm utativ e in t egral domain (with c haracteristic zero) setting w a s considered in [ 343 ]. Some other generalizations to algebraic 13 division algebras and alg ebraic num b er ﬁelds w ere treated in [ 50 , 80 ]. T ao [ 317 ] settled the sum-pro duct problem in arbitrary rings. As w e already men tioned, the sum-pro duct theorem is certainly not true for matrices o ve r F p . Ho w ev er, Helfgo tt [ 183 ] prov es that the theorem is true for A ⊂ S L 2 ( F p ) . In part icular, the set A · A · A is muc h larger than A (more precisely , | A · A · A | > | A | 1+ ε , where ε > 0 is an absolute constant), unless A is con tained in a prop er subgroup. Helfgott’s theorem has fo und sev eral applications, for instance, in some nonlinear sieving pro blems [ 55 ], in the sp ectral theory of Heck e op erators [ 125 ], and in constructing expanders via Cay ley graphs [ 53 ]. Underlying this theorem is the sum-pro duct theorem. V ery recen tly , Kow alski [ 206 ] obtained explicit v ersions of Helfgott’s growth theorem f or S L 2 . Helfgott [ 184 ] prov es his result when A ⊂ S L 3 ( F p ), as w ell. Gill and Helfgo tt [ 133 ] generalized Helfgo tt’s t heorem to S L n ( F p ), when A is small, that is, | A | ≤ p n +1 − δ , for some δ > 0. The study o f grow t h inside solv able subgroups of GL r ( F p ) is done in [ 134 ]. Breuillard, Gr een, a nd T a o [ 67 ], and a lso Pyb er and Szab´ o [ 248 ], indep enden tly and sim ultaneously , generalized Helfgott’s theorem to S L n ( F ), ( n arbitr ary , F ar bitrary ﬁnite ﬁeld), and also to some other simple gro ups, as part of a more general result for groups of b ounded Lie ra nk; see also [ 190 ]. Let us ask do es there exist a ‘sum-division theory’ ? Solymosi [ 295 ] using the concept of multiplic ative en e r gy pro ve d the fo llo wing: If A is a ﬁnite set of p ositiv e real num b ers, then | A + A | 2 | A · A | ≥ | A | 4 4 ⌈ log 2 | A |⌉ . Solymosi’s result also gives | A + A | 2 | A/ A | ≥ | A | 4 4 ⌈ log 2 | A |⌉ . Li and Shen [ 216 ] remo ved the term ⌈ log 2 | A |⌉ in the denominator. In fact, they prov ed the follo wing: If A is a ﬁnite set of p ositiv e real n umbers, then | A + A | 2 | A/ A | ≥ | A | 4 4 , whic h concludes that max {| A + A | , | A/ A |} ≥ | A | 4 / 3 2 . One may ask ab out a ‘diﬀerence-pro duct theory’. The w o rk of Solymosi [ 293 ] considers this t yp e, but the state-of-the-art not only for this type but also for all combinations of addition, m ultiplication, subtraction and division in the case o f complex n um b ers is due to R udnev [ 267 ]. As w e already men tioned, Rudnev [ 266 ] pro ve d max {| A + A | , | A · A |} ≫ | A | 12 / 11 log 4 / 11 | A | , where A ⊂ F ∗ p with | A | < √ p and p large. In Remark 2 of his pap er, Rudnev [ 2 66 ], men tions an in teresting fa ct: “one can replace either one o r b oth the pro duct set A · A with the ratio 14 set A/ A – in whic h case the logarithmic factor disapp ears – and the sumset A + A with the diﬀerence set A − A ”! The sum-pro duct problem can b e applied eﬃcien tly to construct r and omness extr actors [ 19 , 20 , 38 , 41 , 88 , 103 , 18 2 , 346 ]. Inspired b y this f act, w e are g oing to discuss some prop erties and applications of randomness extractors here. First, note that all cryptographic proto cols and in fact, man y pro blems that arise in cryptograph y , algorit hm design, distributed com- puting, and so on, rely completely on randomness and indeed are impossible to solv e without access to it. A r andomn e s s e x tr a c tor is a deterministic p olynomial-time computable algor ithm that computes a function Ext : { 0 , 1 } n → { 0 , 1 } m , with the pro p ert y that for a n y defectiv e source of randomness X satisfying minimal assumptions, Ext ( X ) is close to b eing uniformly distributed. In other w ords, a randomness extractor is an algorithm that transforms a w eak random source in to an almo st unifo rmly random source. Ra ndomness extractors a re in teresting in their o wn righ t as com binatorial ob jects that “app ear random” in many strong w ay s. They fall in to the class of “pseudorandom” ob jects. Pseudor andomness is the theory of eﬃcie ntly generating ob jects that “appear random” ev en though they are construc t ed with lit tle o r no true randomness; see [ 331 , 334 , 335 ] (and also the surv eys [ 27 9 , 28 0 ]). Error correcting co des, hardness ampliﬁers, epsilon biased sets, pseudorandom generators, expander gra phs, and Ramsey graphs are of other suc h ob jects. (Roughly sp eaking, an exp ander is a highly connected sparse ﬁnite graph, i.e., ev ery subset of its v ertices has a lar ge set of neighbors. Expanders ha ve a great deal of seminal applications in many disciplines suc h as computer science and cryptogr aph y; see [ 186 , 22 9 ] for t wo excellen t surv eys on this area and its applications.) Actually , when studying large com bina torial o b jects in a dditiv e com binatorics, a helpful (and e a sier) pro cedure is to decomp ose them in to a ‘structured part’ and a ‘pseudorandom part’. Constructions of randomness extractors hav e b een used to get constructions of communi- cation net works and go o d e xpander graphs [ 76 , 344 ], error corr ecting co des [ 166 , 327 ], crypto- graphic proto cols [ 228 , 333 ], data structures [ 235 ] and samplers [ 349 ]. Randomness extractors are widely used in cryptographic applications (see, e.g., [ 25 , 87 , 99 , 100 , 199 , 20 0 , 218 , 348 ]). This includes applications in construction of pseudorandom generators from one-w a y func- tions, design of cryptographic f unctionalities from noisy a nd weak sources , construction of k ey deriv ation functions, and extracting many priv ate bits ev en when the adv ersary kno ws all exce pt log Ω(1) n o f the n bits [ 251 ] (see also [ 250 ]). They also ha v e remark able applications to quantum cryptograph y , where photons are used b y the randomness extractor to generate secure random bits [ 279 ]. R amsey gr aphs (that is, graphs that ha ve no larg e clique o r indep enden t set) ha v e strong connections with extractors for tw o sources. Using this approac h, Barak et al. [ 21 ] presen ted an explic it R amsey g raph that do es no t ha v e cliques and indep enden t sets of size 2 log o (1) n , and ultimately b eating the F rankl-Wilson construction! 15 Ac kno wl edgements The a uthor w ould lik e to thank Igor Shparlinski for many in v a luable commen ts and for his inspiration throughout the preparation of this surv ey and his une nding encouragemen t. I also thank An tal Balo g, Emman uel Breuillard, Ernie Cro ot, Harald Helfgott, Sergei K on yagin, Liangpan Li, Helger Lipmaa , Dev ansh u P andey , Alain Plagne, L´ aszl´ o Pyb er, Jeﬀrey Shallit, Eman uele Viola, and V an V u for useful commen ts on t his manus cript and/or sending some pap ers to me. Fina lly , I am grateful to the anon ymous referees for their suggestions to impro ve this pap er. References [1] D. 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