Edit: Here is a counterexample that I found with your help: $A = \{1, 2, 4, 5, 9, 12, 13, 17, 21, 24, 25, 29, 33, 37, 40, 41, 45, 49, 53, 56, 57, 61, 65, 69, 72, 73, 77, 81, 85, 88, 89, 93, 97, 101, 104, 105, 109, 113, 117, 120, 121, 125, 129, 133, 136, 137, 141, 145, 149, 152, 153, 157, 161, 165, 168, 169, 173, 177, 181, 184, 185, 189, 193, 197, 200, 201, 205, 209, 213, 216, 217, 221, 225, 229, 232, 233, 237, 241, 245, 248, 249, 253, 257, 261, 264, 265, 269, 273, 277, 280, 281, 285, 289, 293, 296, 297, 301, 305, 309, 312, 313, 317, 321, 325, 328, 329, 333, 337, 341, 344, 345, 349, 353, 357, 360, 361, 365, 369, 373, 376, 377, 381, 385, 389, 392, 393, 397, 401, 405, 408, 409, 413, 417, 421, 424, 425, 429, 433, 437, 440, 441, 445, 449, 453, 456, 457, 461, 465, 469, 472, 473, 477, 481, 485, 488, 489, 493, 497, 501, 504, 505, 509, 513, 517, 520, 521, 525, 529, 533, 536, 537, 541, 545, 549, 552, 553, 557, 561, 565, 568, 569, 573, 577, 581, 584, 585, 589, 593, 597, 600, 601, 605, 609, 613, 616, 617, 621, 625, 629, 632, 633, 637, 641, 645, 648, 649, 653, 657, 661, 664, 665, 669, 673, 677, 680, 681, 685, 689, 693, 696, 697, 701, 705, 709, 712, 713, 717, 721, 725, 728, 729, 733, 737, 741, 744, 745, 749, 753, 757, 760, 761, 765, 769, 773, 776, 777, 781, 785, 789, 792, 793, 797, 801, 805, 809, 813, 817, 820, 821, 825, 828, 829, 830, 848, 4250, 8500, 12750, 17000, 21250, 25500, 29750, 34000, 38250, 42500, 46750, 51000, 55250, 59500, 63750, 68000, 72250, 76500, 80750, 85000, 89250, 93500, 97750, 102000, 106250, 110500, 114750, 119000, 123250, 127500, 131750, 136000, 140250, 144500, 148750, 153000, 157250, 161500, 165750, 170000, 174250, 178500, 182750, 187000, 191250, 195500, 199750, 204000, 208250, 212500, 216750, 221000, 225250, 229500, 233750, 238000, 242250, 246500, 250750, 255000, 259250, 263500, 267750, 272000, 276250, 280500, 284750, 289000, 293250, 297500, 301750, 306000, 310250, 314500, 318750, 323000, 327250, 331500, 335750, 340000, 344250, 348500, 352750, 357000, 361250, 365500, 369750, 374000, 378250, 382500, 386750, 391000, 395250, 399500, 403750, 408000, 412250, 416500, 420750\}$
With this set $A + A - A$ contains 164039 positive and 164834 negative elements.
