INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)

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1 INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li in suitabl short intrvals Our work rlis on a man valu thorm for incomlt Kloostrman sums Introduction Lt b a rim, and lt I, I (0, ) b subintrvals This ar is motivatd by dtrmining conditions on I, I undr which w can nsur th solubility of th congrunc xy (mod ), (x, y) I I An xtnsiv survy of ths modular hyrbolas can b found in th work of Sharlinski [5] From a huristic oint of viw w would xct this congrunc to hav a solution whnvr I, I / Howvr, as highlightd by Hath-Brown [], th bst rsult to dat rquirs that I I 3/ log Th roof rquirs on to stimat incomlt Kloostrman sums S(n, H) = n+h m=n+ m 0 (mod ) for l (Z/lZ), for which th Wil bound yilds lm, S(n, H) ( + log ) / () It has bn conjcturd by Hooly [4] that S(n, H) H / q ε, for any ε > 0, which would nabl on to handl intrvals with I, I /3+ε Howvr such a bound aars to rmain a distant rosct A diffrnt aroach to this roblm involvs considring a squnc of airs of intrvals I (j), I (j), for j J, and to ask whthr thr is a valu of j for which thr is a solution to th congrunc xy (mod ), (x, y) I (j) I (j) () Thr ar som obvious dgnrat cass hr For xaml, if w suos that I (j) = I (j) for all j, and that ths run ovr all intrvals of a givn lngth H, thn w ar mrly asking whthr thr is ositiv intgr h H with th rorty that th congrunc x(x + h) (mod ) has a solution x Z This is quivalnt to dciding whthr th st 00 Mathmatics Subjct Classification L05 (N5) kywords: incomlt Kloostrman sums, modular hyrbolas, Wil s bound

2 T D BROWNING AND A HAYNES {h + 4 : h H} contains a quadratic rsidu modulo Whn H =, thrfor, it is clar that this roblm has a solution for all rims = ± (mod 8) W avoid considrations of this sort by assuming that at last on of our squncs of intrvals is airwis disjoint Th following is our main rsult Thorm Lt H, K > 0 and lt I (j), I (j) (0, ) b subintrvals, for j J, such that I (j) = H and I (j) = K and I (j) I (k) = for all j k Thn thr xists j {,, J} for which () has a solution if J 3 log 4 H K If w tak J = in th thorm thn w rtriv th abov rsult that () is solubl whn HK 3/ log Altrnativly, if w allow a largr valu of J, thn w can gt closr to what would follow on Hooly s hyothsisd bound for S(n, H) Corollary With notation as in Thorm, suos that J /3 Thn thr xists j {,, J} for which () has a solution rovidd that H > /3 and K > /3 (log ) Our roof of Thorm rlis uon a man valu stimat for incomlt Kloostrman sums Ths tys of stimats hav bn studid xtnsivly for multilicativ charactrs, scially in connction with variants of Burgss s bounds (s Hath-Brown [3] and th discussion thrin) Th situation for Kloostrman sums is rlativly undr-dvlod (s Fridlandr and Iwanic [], for xaml) Th rsult w rsnt hr aars to b nw, although many of our tchniqus ar borrowd dirctly from th tratmnt of th analogous multilicativ roblm [3, Thorm ] Th dst art of our argumnt is an aal to Wil s bound for Kloostrman sums W will rov th following rsult in th nxt sction Thorm If I,, I J (0, ) ar disjoint subintrvals, with H/ < I j H for ach j, thn for any l (Z/Z), w hav J ln log H j= n I j Taking J = shows that, u to a constant factor, this rsult includs as a scial cas th bound () for incomlt Kloostrman sums Acknowldgmnts Whil working on this ar TDB was suortd by EPSRC grant EP/E0536/ and AH was suortd by EPSRC grant EP/J0049X/ W ar gratful to Profssor Hath-Brown for svral usful discussions

3 MEAN VALUE THEOREM AND MULTIPLICATIVE INVERSES 3 Proof of Thorm Our starting oint is th following man valu thorm for S(n, H) Lmma For H N and l (Z/Z), w hav S(n, H) H + 8H n= Proof Aftr squaring out th innr sum and intrchanging th ordr of summation, th lft hand sid bcoms H l(n + h n + h ) h,h = n= n h, h (mod ) Using orthogonality of charactrs it is asy to s that th innr sum ovr n is l(x y) = = Hnc S(n, H) = n= x,y (Z/Z) x y h h (mod ) x,y (Z/Z) = a= h,h = ( l(x y) a(x y) + a(h h ) H K(l, a; ) a= a= a(h h ) H h,h = x,y (Z/Z) a(h h ), ) l(x y) + a(x y) whr K(l, a; ) is th usual comlt Kloostrman sum Th contribution from a = is K(l, 0; ) H sinc l Th rmaining contribution has modulus H K(l, a; ) a(h h ) 4 a= h,h = = H, 8 8H, 0< a / 0<a / h,h = min H a(h h ) { H, } a by th Wil bound for th Kloostrman sum and th familiar stimat for a gomtric sris Combining ths contributions, w thrfor arriv at th statmnt of th lmma

4 4 T D BROWNING AND A HAYNES Th rst of th roof of Thorm is takn from th roof of [3, Thorm ], and w includ it only for comltnss W may assum that H 4 in what follows sinc th rsult is trivial othrwis Writ N j for th smallst intgr in I j and suos that N < < N J By saratly considring th odd and thn th vn numbrd intrvals w may assum without loss of gnrality that N j+ N j H for j < J Th starting oint is th obsrvation that J j= n I j ln J max S(N j, h) (3) h H j= For any h H and N j H < n N j w hav that whnc S(N j, h) = S(n, N j n + h) S(n, N j n) k), k H Cauchy s inquality yilds j= S(N j, h) H S(N j, h) 4 H N j H<n N j N j H<n N j k) k H k H k) Taking th max ovr h and thn summing ovr j now givs J max S(N j, h) 4 J k) h H H k H 4 H j= N j H<n N j n= k H k), th last inquality coming from our sacing assumtion W now sk an ur bound for th sum on th right hand sid Lt t b th smallst ositiv intgr with H t, so that in articular H t 4H and t + 4 log H For ach n w choos a ositiv intgr k = k(n) H, with h) = S(n, k) h H By writing k = d D t d, whr D is a collction of intgrs in [0, t], w hav that S(n, k) = d D S(n + v n,d t d, t d ), (4) whr v n,d = D <d d < d

5 MEAN VALUE THEOREM AND MULTIPLICATIVE INVERSES 5 Thn by Cauchy s inquality w dduc that h H h) D S(n + v n,d t d, t d ) d D (t + ) 0 d t 0 v< d S(n + v t d, t d ) Now summing both sids ovr n and alying Lmma w hav that k H k) (t + ) t d + 8 t d n= 0 d t 0 v< ( d ) (t + ) t + 8t H 8 + H log H Sinc H / H w asily comlt th roof of Thorm by combining this with (3) and (4) 3 Proof of Thorm Now w rocd to th roof of our main thorm For ach j th numbr of solutions to () is qual to = l(x y), x I(j) = l= + l= l(x y) = S,j + S,j, say Th total contribution from th S,j trms is J S,j JHK (5) j= Nxt, th standard stimat for a gomtric sris givs S,j = ( ly 0< l / 0< l / l lx ) lx

6 6 T D BROWNING AND A HAYNES Alying Cauchy s inquality and Thorm w dduc that J S,j J lx l j= 0< l / j= J / J lx l 0< l / j= J / (log ) ( log H ) / Undr th conditions of Thorm, it now follows that (5) dominats this quantity, from which th conclusion of th thorm follows Rfrncs [] JB Fridlandr, H Iwanic, Incomlt Kloostrman sums and a divisor roblm Ann of Math (985), [] DR Hath-Brown, Arithmtic alications of Kloostrman sums Niuw Arch Wiskd (000), [3] DR Hath-Brown, Burgss s bounds for charactr sums Submittd, 0 (arxiv:0359) [4] C Hooly, On th gratst rim factor of a cubic olynomial J rin angw Math 303/304 (978), 50 [5] I Sharlinski, Modular hyrbolas Ja J Math, to aar School of Mathmatics, Univrsity of Bristol, Bristol, BS8 TW, UK addrss: tdbrowning@bristolacuk addrss: alanhayns@bristolacuk /

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